#509079

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

729 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)(18,32,31,22)(21,37,39,30)(24,27,40,34)(26,43,35,41)(33,42,44,36);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,20,23,25)(16,26,29,35)(18,27,31,34)(21,36,39,42)(22,40,32,24)(28,41,38,43)(30,33,37,44);(2,4)(13,16,17,29)(14,21,23,39)(15,26,19,35)(18,33,31,44)(20,36,25,42)(22,38,32,28)(24,41,40,43)(27,37,34,30);(6,10)(7,12)(14,22)(15,19)(18,34)(20,24)(21,38)(23,32)(25,40)(26,35)(27,31)(28,39)(30,33)(36,41)(37,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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=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, 416275);