#504132

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411328);