#511527

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=i,d^2=ef,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, 418723);