#6501006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,43)(30,38)(32,42)(33,39)(34,41)(35,40)(37,44)(45,47)(46,48);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,27,38)(20,29,34,45)(21,31,22,30)(25,37,41,48)(26,42,36,28)(33,46,35,44)(39,43,40,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,23,34)(18,25,27,41)(19,29,32,45)(21,35,22,33)(24,37,38,48)(26,40,36,39)(28,43,42,47)(30,46,31,44);(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,42)(25,39,41,40)(29,44,45,46)(37,47,48,43)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,g^c=gi,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, 6408202);