#6757227

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,e^2=i,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, 6664423);