#503994

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

598 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)(7,11)(8,9)(10,12)(13,14)(15,29)(16,34)(17,37)(18,26)(19,25)(20,27)(21,40)(22,30)(23,32)(24,36)(28,38)(31,44)(33,42)(35,43)(39,41);(1,2)(3,4)(6,9)(10,11)(13,15,19,31,21,33,35,16,20,32,39,36,38,17,18,30)(14,22,26,37,28,24,41,23,27,34,43,42,40,44,25,29);(2,4)(13,16,35,17,38,31,19,32,20,30,18,33,21,36,39,15)(14,23,41,44,40,37,26,34,27,29,25,24,28,42,43,22);(5,7,8,12)(6,10,9,11)(13,17,39,16,21,15,18,36,20,33,19,30,38,32,35,31)(14,24,43,29,28,34,25,37,27,44,26,23,40,22,41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411190);