#6464730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,39)(30,44)(32,45)(33,40)(34,48)(35,42)(36,43)(38,47)(41,46);(9,10)(11,15)(12,13)(14,16)(17,19,22,31,23,32,20,29)(18,24,25,28,27,39,37,45)(21,34,35,46,36,30,33,47)(26,41,40,48,43,38,42,44);(2,5)(6,8)(10,13)(14,15)(17,20,23,22)(18,25,27,37)(19,29,32,31)(21,35,36,33)(24,28,39,45)(26,42,43,40)(30,47,34,46)(38,48,41,44);(9,11,12,16)(10,14,13,15)(17,21,23,36)(18,26,27,43)(19,30,32,34)(20,33,22,35)(24,38,39,41)(25,40,37,42)(28,44,45,48)(29,46,31,47);(1,3,4,7)(2,6,5,8)(21,36)(26,43)(30,34)(33,35)(38,41)(40,42)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,d^c=di,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, 6371926);