#6842116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,31)(21,37)(22,27)(23,28)(25,34)(26,38)(29,45)(30,40)(32,44)(33,42)(35,41)(36,48)(39,47)(43,46);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,40)(20,32,34,35)(21,23,38,33)(25,41,31,44)(26,42,37,28)(29,46,47,36)(39,48,45,43);(2,6)(7,8)(10,14)(11,16)(17,20,33,46,22,34,23,36)(18,25,42,48,27,31,28,43)(19,29,38,35,30,47,21,32)(24,39,26,44,40,45,37,41);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(20,34)(25,31)(29,47)(32,35)(36,46)(39,45)(41,44)(43,48);(1,4,5,3)(2,8,6,7)(17,21,23,30,22,38,33,19)(18,26,28,24,27,37,42,40)(20,35,36,29,34,32,46,47)(25,41,43,45,31,44,48,39)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6749312);