#72251914

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	#2623192
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#3003	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3264 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=fi,d^2=fi,e^2=i,g^2=j,h^2=j,b^a=bf,c^b=cfghj,d^a=dfgi,d^b=df,e^a=ei,e^b=egh,e^c=ehi,e^d=eghi,f^c=fj,f^e=fj,g^d=gj,h^a=hj,h^c=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251956
maximal normal subgroup of	#72251931, #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, 1, F.7);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);