#6993242

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(4,6)(5,8)(17,18,23,37,21,26,22,27)(19,28,38,40,30,24,31,44)(20,32,36,25,34,43,35,42)(29,45,47,41,33,48,46,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,38)(25,41)(26,28)(27,40)(29,34)(32,45)(35,47)(36,46)(37,44)(39,42)(43,48);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20,21,34)(18,25,26,42)(19,29,30,33)(22,36,23,35)(24,39,28,41)(27,43,37,32)(31,47,38,46)(40,45,44,48);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(18,26)(19,30)(25,42)(27,37)(29,33)(31,38)(32,43)(46,47);(9,13,14,12)(10,15,11,16)(18,26)(24,28)(25,42)(27,37)(32,43)(39,41)(40,44)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900438);