#72251924

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#4649504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#417597	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3407 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(5,22)(8,29)(24,31)(25,28);(1,2)(3,8,16,29)(4,18,20,32)(5,23)(6,11)(7,12)(9,21,15,27)(10,28)(13,30)(14,25)(17,24,26,31)(19,22);(1,3,13,17,7,16,6,26)(2,8,30,24,12,29,11,31)(4,19,21,14,20,23,27,10)(5,15,28,32,22,9,25,18);(1,4,13,21,7,20,6,27)(2,9,30,18,12,15,11,32)(3,14,26,19,16,10,17,23)(5,24,25,8,22,31,28,29);(1,5,7,22)(2,10,12,14)(3,15,16,9)(4,8,20,29)(6,28,13,25)(11,19,30,23)(17,18,26,32)(21,31,27,24)

Polycyclic presentation:

2^10|b^2=g,c^2=ghj,d^2=fghj,e^2=j,h^2=j,b^a=bf,c^a=cfi,c^b=cgij,d^b=dfh,d^c=dghij,e^a=eij,e^b=eghij,e^c=eij,e^d=eh,f^c=fj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,h^d=hj,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251940, #72251956
maximal normal subgroup of	#72251931, #72251940, #72251956

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);