#6463580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,27)(19,29,32,24)(20,34,36,48)(21,26,33,41)(23,39,31,28)(25,42,44,37)(30,45,47,43)(35,46,38,40);(2,5)(6,8)(11,12)(14,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,40)(26,39)(27,29)(28,41)(34,43)(35,37)(36,47)(38,42)(44,46)(45,48);(9,11,13,12)(10,14,15,16)(17,20,31,42,22,36,23,37)(18,25,39,48,27,44,28,34)(19,30,21,38,32,47,33,35)(24,40,26,45,29,46,41,43);(9,12,13,11)(10,14,15,16)(17,21,22,33)(18,26,27,41)(19,31,32,23)(20,35,36,38)(24,39,29,28)(25,43,44,45)(30,37,47,42)(34,46,48,40);(1,3,4,7)(2,6,5,8)(19,32)(20,36)(21,33)(24,29)(25,44)(26,41)(34,48)(37,42)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370776);