#72251685

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2046711
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

4107 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,d^2=i,h^2=j,c^a=cfj,c^b=cg,d^a=dgj,d^b=dghij,d^c=dgj,e^c=eghij,e^d=eij,f^d=fj,g^c=gj,h^b=hj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251715, #72251900, #72251931
maximal normal subgroup of	#72251715, #72251900, #72251931

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.6);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);