#510052

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

509 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417248);