#504861

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

623 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)(13,14,18,23)(15,25,16,30)(17,33,35,37)(19,36,34,24)(20,31,21,28)(22,26,40,27)(29,44,32,41)(38,43,39,42);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,26,42,19,29,17,28,18,16,27,43,34,32,35,31)(14,20,37,44,24,38,22,25,23,21,33,41,36,39,40,30);(2,4)(6,10)(9,11)(15,16)(20,21)(25,30)(28,31)(29,32)(38,39)(41,44)(42,43);(6,10)(9,11)(13,16,26,43,19,32,17,31,18,15,27,42,34,29,35,28)(14,21,37,41,24,39,22,30,23,20,33,44,36,38,40,25)

Polycyclic presentation:

2^9|a^2=i,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^c=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, 412057);