#511339

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,7,8,6)(9,10,13,15)(11,14,12,16)(17,18,20,24)(19,23,28,36)(21,25,31,39)(22,26,32,40)(27,41,45,35)(29,37,46,48)(30,38,34,44)(33,43,47,42);(2,5)(3,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,27)(22,34)(24,36)(25,41)(26,44)(29,33)(30,32)(31,45)(35,39)(37,43)(38,40)(42,48)(46,47);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(21,33)(22,32)(23,35)(25,42)(28,45)(29,30)(31,47)(34,46)(36,41)(37,44)(38,48)(39,43);(1,3,4,8)(2,6,5,7)(10,15)(14,16)(18,24)(23,36)(25,39)(26,40)(35,41)(37,48)(38,44)(42,43)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=fi,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=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, 418535);