#6771057

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 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)(19,27)(20,31)(21,35)(22,28)(23,24)(25,29)(26,37)(30,38)(32,45)(33,40)(34,42)(36,41)(39,46)(43,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,28,27)(20,29,34,33)(21,36,37,30)(25,31,40,42)(26,41,35,38)(32,46,47,43)(39,45,48,44);(11,16)(13,15)(17,20,22,34)(18,25,28,40)(19,29,23,33)(21,32,37,47)(24,31,27,42)(26,39,35,48)(30,43,36,46)(38,44,41,45);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,37)(18,26,28,35)(19,30,23,36)(20,32,34,47)(24,38,27,41)(25,39,40,48)(29,43,33,46)(31,44,42,45);(2,5)(6,8)(17,19,22,23)(18,27,28,24)(20,33,34,29)(21,36,37,30)(25,31,40,42)(26,38,35,41)(32,43,47,46)(39,45,48,44)

Polycyclic presentation:

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

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, 6678253);