#503763

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=fi,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, 410959);