#6611945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,27,23,28,21,26)(19,29,31,38,32,42,36,24)(20,25,35,41,33,39,34,40)(30,43,45,47,44,48,46,37);(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,42)(28,38)(33,44)(34,45)(35,46)(39,47)(40,43)(41,48);(2,5)(6,8)(10,15)(11,12)(17,20,23,33)(18,25,28,39)(19,30,32,44)(21,34,22,35)(24,37,38,47)(26,40,27,41)(29,43,42,48)(31,45,36,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(20,33)(25,39)(30,44)(34,35)(37,47)(40,41)(43,48)(45,46);(11,12)(14,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519141);