#72266830

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#449501
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#6499892	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4783 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)(3,11)(4,15)(5,9)(6,10)(7,13)(8,18)(12,20)(14,21)(16,29)(17,23)(19,24)(22,31)(25,27)(26,28)(30,32);(7,11)(20,25)(21,26)(22,29);(1,3,5,14)(2,7,9,21)(4,16,17,12)(6,13,19,28)(8,22,23,25)(10,11,24,26)(15,29,30,20)(18,31,32,27);(1,4)(2,8,10,15)(3,12)(5,17)(6,18)(7,20,11,25)(9,23,24,30)(13,27)(14,16)(19,32)(21,29,26,22)(28,31)

Polycyclic presentation:

2^10|c^2=f,d^2=h,b^a=be,c^a=cg,c^b=ch,d^a=dgi,d^c=dfh,e^c=ej,f^a=fi,g^c=gi,g^f=gj,h^a=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266831, #72266950, #72309896
maximal normal subgroup of	#72266831, #72266950, #72309896

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