#6996521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,21)(17,20)(18,23)(22,29)(24,31)(25,30)(26,33)(27,36)(28,34)(32,39)(35,40)(37,43)(38,41)(42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,23,33)(16,24,27,37)(17,25,29,39)(20,30,22,32)(21,31,36,43)(28,38,40,44)(34,41,35,42);(5,7)(6,9)(8,12)(10,11)(14,20)(16,27)(17,29)(19,30)(21,35)(22,23)(24,37)(25,39)(31,42)(32,33)(34,36)(41,43);(1,3,4,2)(7,12)(9,11)(13,16,18,27)(14,21,23,36)(15,24,26,37)(17,28,29,40)(19,31,33,43)(20,34,22,35)(25,38,39,44)(30,41,32,42);(6,10)(9,11)(16,27)(21,36)(24,37)(28,40)(31,43)(34,35)(38,44)(41,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903717);