#6491950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(11,15)(14,16)(17,18)(19,28)(20,32)(21,26)(22,27)(23,36)(24,31)(25,33)(29,44)(30,40)(34,41)(35,39)(37,42)(38,45)(43,47)(46,48);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,28)(20,29,34,46)(21,35,23,30)(25,38,42,43)(26,40,36,39)(32,48,41,44)(33,47,37,45);(10,13)(14,16)(17,20,22,34)(18,25,27,42)(19,29,31,46)(21,33,23,37)(24,38,28,43)(26,41,36,32)(30,45,35,47)(39,48,40,44);(10,13)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,35)(20,33,34,37)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44);(9,11,12,15)(10,14,13,16)(19,31)(24,28)(29,46)(30,35)(38,43)(39,40)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6399146);