#6764503

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,33,36,44)(21,28,23,26)(25,38,42,35)(30,45,34,48)(31,40,37,43)(39,47,41,46);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,34)(21,37)(22,32)(23,31)(25,41)(26,43)(27,29)(28,40)(30,36)(33,45)(35,46)(38,47)(39,42)(44,48);(10,14)(11,15)(17,20,21,38,22,36,23,35)(18,25,26,44,27,42,28,33)(19,30,37,46,32,34,31,47)(24,39,43,45,29,41,40,48);(2,5)(6,8)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,40,29,43)(25,33,42,44)(30,46,34,47)(39,45,41,48);(1,3,4,7)(2,6,5,8)(11,15)(13,16)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

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, 6671699);