#504774

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

583 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)(13,14,16,22,31,39,41,21,18,24,32,37,17,23,30,20)(15,25,26,40,33,44,43,36,28,34,42,35,27,38,29,19);(1,2)(3,4)(6,9)(10,11)(13,15,18,28)(14,19,24,36)(16,29,32,43)(17,33,31,27)(20,25,21,34)(22,38,37,44)(23,40,39,35)(26,41,42,30);(2,4)(14,20)(16,30)(17,31)(19,25)(21,24)(22,23)(26,29)(27,33)(32,41)(34,36)(35,44)(37,39)(38,40)(42,43);(5,7,8,12)(6,10,9,11)(14,21)(16,30)(17,31)(19,34)(20,24)(22,39)(23,37)(25,36)(26,29)(27,33)(32,41)(35,38)(40,44)(42,43)

Polycyclic presentation:

2^9|a^2=e,b^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, 411970);