#499463

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	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

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406659);