#505696

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	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

550 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)(13,14,19,24)(15,25,28,43)(16,29,32,42)(17,33,30,44)(18,35,34,23)(20,36,39,27)(21,40,38,31)(22,41,37,26);(5,7)(6,9)(8,11)(10,12)(13,15,26,31,18,16,30,36,19,28,41,40,34,32,17,27)(14,20,33,42,23,38,37,43,24,39,44,29,35,21,22,25);(2,4)(7,11)(9,12);(13,16,26,36,18,28,30,40,19,32,41,27,34,15,17,31)(14,21,33,25,23,20,37,42,24,38,44,43,35,39,22,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,d^2=eh,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, 412892);