#6682476

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2290 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

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, 6589672);