#6407728

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=dfi,d^c=di,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314924);