#504780

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi,e^d=eh,f^d=fh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411976);