#504756

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 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)(13,14,16,21,30,38,41,34,18,23,31,33,17,22,28,19)(15,24,25,39,32,44,43,36,27,40,42,35,26,37,29,20);(1,2)(3,4)(6,9)(10,11)(14,19)(16,28)(17,30)(20,24)(21,22)(23,34)(25,29)(26,32)(31,41)(33,38)(35,44)(36,40)(37,39)(42,43);(2,4)(13,15,18,27)(14,20,23,36)(16,29,31,43)(17,32,30,26)(19,24,34,40)(21,37,33,44)(22,39,38,35)(25,41,42,28);(5,7,8,12)(6,10,9,11)(14,19)(16,28)(17,30)(20,24)(21,22)(23,34)(25,29)(26,32)(31,41)(33,38)(35,44)(36,40)(37,39)(42,43)

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411952);