#6940264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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

1535 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,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,d^c=dfh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6847460);