#511414

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

464 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)(11,15)(12,14)(13,16)(17,18,21,23)(19,27,29,24)(20,31,33,25)(22,26,35,38)(28,39,45,43)(30,44,41,36)(32,40,48,46)(34,47,42,37);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,24,23,27)(20,32,33,48)(22,36,35,44)(25,40,31,46)(26,41,38,30)(28,42,45,34)(37,39,47,43);(10,14)(13,15)(17,20,21,33)(18,25,23,31)(19,28,29,45)(22,37,35,47)(24,39,27,43)(26,42,38,34)(30,46,41,40)(32,44,48,36);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(19,29)(24,27)(28,45)(30,41)(32,48)(36,44)(39,43)(40,46)

Polycyclic presentation:

2^9|a^2=efi,b^2=i,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, 418610);