#6771473

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

1537 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)(19,24)(20,31)(21,26)(22,35)(23,27)(25,32)(28,43)(29,38)(30,39)(33,41)(34,40)(36,42)(37,46)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,33,44)(21,34,22,30)(25,37,42,47)(26,40,35,39)(31,43,41,48)(32,46,36,45);(3,8)(5,7)(10,13)(14,16)(17,20,23,33)(18,25,27,42)(19,28,29,44)(21,32,22,36)(24,37,38,47)(26,41,35,31)(30,45,34,46)(39,43,40,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,44)(30,34)(37,47)(39,40)(43,48)(45,46);(10,13)(14,16)(17,21,23,22)(18,26,27,35)(19,30,29,34)(20,32,33,36)(24,39,38,40)(25,41,42,31)(28,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,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,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, 6678669);