#6488889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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

1345 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)(11,14)(12,15)(13,16)(17,18)(19,27)(20,24)(21,33)(22,25)(23,30)(26,34)(28,39)(29,40)(31,37)(32,45)(35,42)(36,43)(38,44)(41,47)(46,48);(11,12)(14,15)(17,19,21,30,22,28,34,42)(18,23,26,27,25,35,33,39)(20,29,32,43,31,41,44,48)(24,36,38,40,37,46,45,47);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(19,28)(20,31)(23,35)(24,37)(27,39)(30,42)(32,44)(38,45);(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,29)(21,32)(22,31)(23,36)(25,37)(26,38)(27,40)(28,41)(30,43)(33,45)(34,44)(35,46)(39,47)(42,48);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(18,25)(20,31)(23,35)(26,33)(27,39)(29,41)(32,44)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396085);