#499467

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

620 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:

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

Polycyclic presentation:

2^9|a^2=fi,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, 406663);