#506327

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 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)(5,6,8,9)(7,11,12,10);(6,9)(10,11)(13,14,16,21,28,35,41,39,18,20,30,34,17,22,29,37)(15,23,24,38,31,44,43,32,26,40,42,33,25,36,27,19);(2,4)(13,15,18,26)(14,19,20,32)(16,27,30,43)(17,31,28,25)(21,36,34,44)(22,38,35,33)(23,39,40,37)(24,41,42,29);(5,7,8,12)(6,10,9,11)(14,20)(19,32)(21,34)(22,35)(23,40)(33,38)(36,44)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413523);