#6940266

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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(17,18,23,27)(19,24,29,38)(20,31,34,45)(21,26,22,36)(25,32,41,33)(28,42,43,48)(30,39,35,40)(37,46,47,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,35)(22,30)(23,29)(25,37)(26,40)(27,38)(31,42)(32,46)(33,44)(34,43)(36,39)(41,47)(45,48);(3,8)(5,7)(10,13)(11,15)(17,20,22,33,23,34,21,32)(18,25,26,31,27,41,36,45)(19,28,30,44,29,43,35,46)(24,37,40,42,38,47,39,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,43)(30,35)(37,47)(39,40)(42,48)(44,46);(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,32,34,33)(24,39,38,40)(25,31,41,45)(28,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fh,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, 6847462);