#508395

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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)(7,10)(8,9)(11,12)(13,14,17,21,30,35,41,40,18,20,31,34,19,22,32,37)(15,16,24,26,38,33,44,43,23,27,39,42,25,28,36,29);(1,2)(3,4)(6,9)(10,12)(13,15)(14,16)(17,24)(18,23)(19,25)(20,27)(21,26)(22,28)(29,37)(30,38)(31,39)(32,36)(33,35)(34,42)(40,43)(41,44);(2,4)(7,11)(10,12)(13,16,18,27)(14,15,20,23)(17,29,31,43)(19,33,30,28)(21,36,34,44)(22,38,35,25)(24,40,39,37)(26,41,42,32);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(21,34)(22,35)(24,39)(25,38)(36,44)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415591);