#508943

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=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, 416139);