#6473591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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

1476 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)(17,18,23,28)(19,29,21,25)(20,32,34,43)(22,27,24,38)(26,37,44,33)(30,45,35,48)(31,41,36,40)(39,47,42,46);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,21)(18,25,28,29)(20,30,34,35)(22,36,24,31)(26,39,44,42)(27,41,38,40)(32,48,43,45)(33,47,37,46);(11,15)(13,16)(17,20,22,37,23,34,24,33)(18,26,38,43,28,44,27,32)(19,30,36,46,21,35,31,47)(25,39,40,45,29,42,41,48);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,23,19)(18,25,28,29)(20,30,34,35)(22,31,24,36)(26,42,44,39)(27,41,38,40)(32,45,43,48)(33,47,37,46);(10,14)(13,16)(17,22,23,24)(18,27,28,38)(19,31,21,36)(20,33,34,37)(25,40,29,41)(26,43,44,32)(30,46,35,47)(39,48,42,45)

Polycyclic presentation:

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

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, 6380787);