#6483525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=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, 6390721);