#72324275

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=g,e^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=dgj,e^a=ei,e^c=eh,f^b=fhij,f^d=fi,f^e=fj,g^a=gi,h^a=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72324276, #72324332, #72324342
maximal normal subgroup of	#72324276, #72324332, #72324342

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