#510041

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

512 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)(13,14,19,33,31,40,30,21,18,24,16,22,17,23,35,38)(15,25,42,36,32,44,29,20,28,43,26,41,27,39,34,37);(1,2)(3,4)(5,7,9,12)(6,10,11,8);(2,4)(13,15,18,28)(14,20,24,37)(16,29,19,34)(17,32,31,27)(21,25,38,43)(22,39,33,44)(23,41,40,36)(26,35,42,30);(5,8,9,10)(6,7,11,12)(14,21)(16,30)(17,31)(19,35)(20,25)(22,23)(24,38)(26,29)(27,32)(33,40)(34,42)(36,44)(37,43)(39,41)

Polycyclic presentation:

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

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, 417237);