#6408074

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315270);