#6459930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

1652 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)(17,18,22,26)(19,28,21,24)(20,31,34,47)(23,37,32,27)(25,40,41,35)(29,44,33,46)(30,39,36,43)(38,45,42,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,21)(18,24,26,28)(20,29,34,33)(23,30,32,36)(25,38,41,42)(27,39,37,43)(31,46,47,44)(35,45,40,48);(10,14)(11,15)(17,20,32,40,22,34,23,35)(18,25,37,47,26,41,27,31)(19,29,36,48,21,33,30,45)(24,38,43,44,28,42,39,46);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,22,19)(18,24,26,28)(20,33,34,29)(23,36,32,30)(25,38,41,42)(27,39,37,43)(31,46,47,44)(35,48,40,45);(19,21)(24,28)(29,33)(30,36)(38,42)(39,43)(44,46)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367126);