#506541

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

795 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)(7,11)(8,10)(9,12)(13,14,18,23,33,31,34,35,20,16,17,22,19,24,36,32)(15,21,28,39,42,44,43,25,26,30,27,38,29,40,37,41);(5,7,8,12)(6,9,10,11)(13,15,17,27,33,42,36,37,20,26,18,28,19,29,34,43)(14,21,22,38,31,44,32,41,16,30,23,39,24,40,35,25);(2,4)(13,16,20,14)(15,25,26,41)(17,32,18,35)(19,24,33,31)(21,37,30,43)(22,34,23,36)(27,40,28,44)(29,38,42,39);(6,10)(9,11)(15,26)(21,30)(25,41)(27,28)(29,42)(37,43)(38,39)(40,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,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, 413737);