#6771010

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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6678206);