#6692951

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(9,10,13,15)(11,14,12,16)(18,24)(19,28)(26,36)(27,39)(30,37)(31,43)(33,42)(45,47);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,29)(20,26)(21,33)(22,24)(23,36)(25,30)(27,34)(28,40)(31,46)(32,39)(35,42)(37,38)(41,43)(44,45)(47,48);(9,12,13,11)(10,14,15,16)(17,19,22,28)(18,25,24,38)(20,30,23,37)(21,31,35,43)(26,40,36,29)(27,41,39,46)(32,45,34,47)(33,48,42,44);(2,6)(5,7)(17,20,22,23)(18,26,24,36)(19,30,28,37)(21,34,35,32)(25,40,38,29)(27,42,39,33)(31,47,43,45)(41,44,46,48);(3,8)(5,7)(17,21,22,35)(18,27,24,39)(19,31,28,43)(20,32,23,34)(25,41,38,46)(26,33,36,42)(29,44,40,48)(30,45,37,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6600147);