#504761

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

574 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=efhi,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, 411957);