#6851629

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6758825);