#72226366

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#115133
commutator	C₂⁶	C₂²×C₄
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	#6437621	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=d,b^a=bg,c^a=cf,c^b=ce,d^a=dj,d^b=dh,e^a=ei,e^c=eh,f^b=fj,f^c=fj,g^c=gij,g^d=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238352, #72255779, #72255888, …
maximal normal subgroup of	#72238352, #72255779, #72255888, …
maximal quotient of	#72226367, #72266334

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