#6873818

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1520 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)(17,18,22,28,23,26,21,27)(19,24,32,39,30,40,31,38)(20,33,34,41,35,42,36,25)(29,43,45,47,44,48,46,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,37)(26,40)(27,38)(28,39)(33,43)(34,45)(35,44)(36,46)(41,47)(42,48);(3,8)(5,7)(10,14)(15,16)(17,20,23,35)(18,25,26,41)(19,29,30,44)(21,34,22,36)(24,37,40,47)(27,33,28,42)(31,45,32,46)(38,43,39,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,26)(19,30)(25,41)(27,28)(29,44)(31,32)(33,42)(45,46);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,26,28)(19,31,30,32)(20,34,35,36)(24,38,40,39)(25,33,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6781014);