#6803406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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

1367 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

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, 6710602);