#6745457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#35659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2303 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,31)(21,34)(22,28)(23,24)(25,29)(26,36)(30,38)(32,45)(33,40)(35,42)(37,41)(39,46)(43,48)(44,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,23)(18,24,28,27)(20,29,33,37)(21,30,36,35)(25,31,41,40)(26,38,34,42)(32,43,47,46)(39,44,48,45);(11,15)(13,16)(17,20,22,33)(18,25,28,41)(19,29,23,37)(21,32,36,47)(24,31,27,40)(26,39,34,48)(30,43,35,46)(38,44,42,45);(2,6)(5,7)(17,21)(18,26)(19,30)(20,32)(22,36)(23,35)(24,38)(25,39)(27,42)(28,34)(29,43)(31,44)(33,47)(37,46)(40,45)(41,48);(3,8)(5,7)(11,15)(13,16)(17,19,22,23)(18,27,28,24)(20,29,33,37)(21,35,36,30)(25,40,41,31)(26,38,34,42)(32,46,47,43)(39,44,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652653);