#6460123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,40)(20,31,32,45)(21,26,22,36)(25,33,41,34)(28,42,43,48)(29,38,35,39)(37,46,47,44);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,39)(27,40)(31,42)(32,43)(33,46)(34,44)(36,38)(41,47)(45,48);(10,14)(11,15)(17,20,21,33,23,32,22,34)(18,25,36,45,27,41,26,31)(19,28,35,46,30,43,29,44)(24,37,38,48,40,47,39,42);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(20,32)(25,41)(28,43)(31,45)(33,34)(37,47)(42,48)(44,46);(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,33,32,34)(24,38,40,39)(25,31,41,45)(28,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di

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, 6367319);