#505688

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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)(5,6,8,10)(7,9,11,12)(13,14)(15,25)(16,30)(17,33)(18,36)(19,24)(20,27)(21,32)(22,26)(23,34)(28,44)(29,41)(31,37)(35,39)(38,43)(40,42);(5,7)(6,9)(8,11)(10,12)(13,15,26,42,18,28,35,43,19,29,31,32,34,16,17,27)(14,20,33,30,23,21,37,41,24,38,39,44,36,40,22,25);(2,4)(7,11)(9,12);(13,16,31,43,18,15,17,32,19,28,26,27,34,29,35,42)(14,21,39,25,23,38,22,30,24,40,33,41,36,20,37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412884);