#6471713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,26)(21,27)(22,24)(23,35)(25,30)(29,43)(31,40)(32,41)(33,38)(34,42)(36,37)(39,46)(44,47)(45,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,37)(26,38)(27,35)(28,40)(39,47)(41,42)(43,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19)(18,25)(20,29)(21,30)(22,31)(23,36)(24,37)(26,39)(27,40)(28,35)(32,44)(33,45)(34,46)(38,47)(41,43)(42,48);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20,22,33)(18,26,24,38)(19,29,31,45)(21,34,23,32)(25,39,37,47)(27,42,35,41)(28,43,40,48)(30,46,36,44);(10,14)(13,16)(17,21,22,23)(18,27,24,35)(19,30,31,36)(20,32,33,34)(25,40,37,28)(26,41,38,42)(29,44,45,46)(39,43,47,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,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,g^c=gi,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, 6378909);