#72334818

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	#1168770
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#2021	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3042 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)(3,11,14,7)(4,15,17,8)(5,9)(6,10)(12,19)(13,20,28,22)(16,21,29,23)(18,24)(25,27)(26,30)(31,32);(1,3)(2,7)(4,12)(5,13,18,28)(6,14)(8,19)(9,22,24,20)(10,11)(15,25)(16,26,29,32)(17,27)(21,31,23,30);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,23)(10,15)(11,25)(13,26)(14,27)(18,29)(20,31)(21,24)(22,30)(28,32);(1,5,6,18)(2,9,10,24)(3,13)(4,16)(7,20)(8,21)(11,22)(12,26,27,32)(14,28)(15,23)(17,29)(19,30,25,31);(4,17)(5,18)(12,27)(13,28)

Polycyclic presentation:

2^10|a^2=h,b^2=f,d^2=hj,b^a=bi,c^a=ci,d^a=dh,d^b=dg,d^c=dfij,e^a=eghi,e^c=eij,e^d=eij,f^d=fj,g^a=gj,g^b=gj,g^d=gj,h^b=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334823, #72334833, #72334861, …
maximal normal subgroup of	#72334823, #72334833, #72334861, …

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