#72251699

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1087286
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4639 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)(13,14)(21,25)(28,31);(1,2)(3,7)(4,15,17,32)(5,19)(6,10)(8,24,26,18)(9,20)(11,23)(12,28,16,31)(13,22)(14,27)(21,29,25,30);(1,3,5,13,6,11,20,14)(2,7,9,27,10,23,19,22)(4,16,24,30,17,12,18,29)(8,25,32,28,26,21,15,31);(1,4)(2,8,10,26)(3,12)(5,18)(6,17)(7,21,23,25)(9,15,19,32)(11,16)(13,30)(14,29)(20,24)(22,31,27,28);(3,13)(4,17)(5,20)(7,22)(9,19)(11,14)(12,29)(15,32)(16,30)(21,31)(23,27)(25,28)

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,d^2=ij,h^2=j,b^a=bg,c^a=cfi,c^b=cgj,d^a=dg,d^b=dghi,d^c=dg,e^c=eghi,e^d=ei,f^d=fj,g^c=gj,h^b=hj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251715, #72251954, #72251959
maximal normal subgroup of	#72251715, #72251954, #72251959

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