#6682291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

2276 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589487);