#509094

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

723 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)(8,10)(9,12)(13,14,17,23)(15,25,19,20)(16,28,29,38)(18,21,31,37)(22,30,39,42)(24,27,35,32)(26,40,34,43)(33,36,44,41);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,20,23,25)(16,26,29,34)(18,27,31,32)(21,24,37,35)(22,36,39,41)(28,43,38,40)(30,33,42,44);(2,4)(14,21)(15,19)(18,32)(20,35)(22,38)(23,37)(24,25)(26,34)(27,31)(28,39)(30,33)(36,43)(40,41)(42,44);(6,10)(7,12)(13,16,17,29)(14,22,23,39)(15,26,19,34)(18,33,31,44)(20,36,25,41)(21,38,37,28)(24,40,35,43)(27,42,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,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, 416290);