#6801493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708689);