#6637465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,29,31,24)(20,33,35,42)(21,28,23,27)(25,37,40,34)(30,44,45,48)(32,39,36,41)(38,47,43,46);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,41)(28,39)(33,48)(34,47)(35,45)(37,46)(40,43)(42,44);(11,15)(13,16)(17,20,23,34,22,35,21,37)(18,25,28,33,26,40,27,42)(19,30,32,47,31,45,36,46)(24,38,39,48,29,43,41,44);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,26)(19,31)(25,40)(27,28)(30,45)(32,36)(33,42)(46,47);(2,5)(6,7)(10,14)(13,16)(17,21,22,23)(18,27,26,28)(19,32,31,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6544661);