#6408080

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1118 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)(33,34,37,36,38,40,45,44)(35,39,43,42,41,46,48,47);(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);(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)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,15,10,16)(18,24)(22,31)(23,27)(30,32)(35,41)(39,46)(42,47)(43,48);(1,5,6,4)(2,8,9,13)(3,12,11,14)(7,16,10,15)(33,36,37,40,38,44,45,34)(35,42,43,46,41,47,48,39)

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,e^a=ei,c^b=cfg,d^b=df,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315276);