#6508232

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415428);