#504865

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	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

626 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)(7,11)(8,10)(9,12)(13,14)(15,23)(16,29)(17,20)(18,32)(19,26)(21,25)(22,30)(24,36)(27,37)(28,44)(31,33)(34,43)(35,40)(38,41)(39,42);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,24,40,18,28,16,26,17,27,25,41,30,42,31,43)(14,19,33,44,22,35,21,23,20,34,29,39,32,38,36,37);(2,4)(6,10)(9,11)(14,20)(19,34)(21,36)(22,32)(23,37)(29,33)(35,38)(39,44);(6,10)(9,11)(13,15,25,41,18,28,31,43,17,27,24,40,30,42,16,26)(14,19,29,39,22,35,36,37,20,34,33,44,32,38,21,23)

Polycyclic presentation:

2^9|b^2=ehi,d^2=efhi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di

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, 412061);