#6572488

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(17,18,23,26)(19,24,30,38)(20,31,33,42)(21,27,22,35)(25,36,41,32)(28,43,46,48)(29,39,34,40)(37,44,47,45);(2,5)(6,8)(9,10,14,13)(11,16,12,15)(17,19,23,30)(18,24,26,38)(20,28,33,46)(21,34,22,29)(25,37,41,47)(27,40,35,39)(31,43,42,48)(32,45,36,44);(9,11,14,12)(10,15,13,16)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,46)(31,35)(34,45)(38,47)(39,43)(40,48);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,13,16)(18,26)(24,38)(25,41)(27,35)(31,42)(37,47)(39,40)(43,48);(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,27,26,35)(19,29,30,34)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479684);