#6693019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18)(19,29)(20,26)(21,27)(22,24)(23,28)(25,32)(30,38)(31,43)(33,41)(34,42)(35,37)(36,39)(40,45)(44,47)(46,48);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(18,24)(19,30)(20,23)(21,34)(25,38)(27,41)(29,39)(31,45)(32,36)(33,35)(37,42)(47,48);(9,12,13,11)(10,14,15,16)(17,19)(18,25)(20,30)(21,31)(22,32)(23,36)(24,29)(26,39)(27,40)(28,38)(33,44)(34,46)(35,45)(37,43)(41,48)(42,47);(2,6)(5,7)(17,20,22,23)(18,26,24,28)(19,30,32,36)(21,34,35,33)(25,39,29,38)(27,42,37,41)(31,46,45,44)(40,47,43,48);(3,8)(5,7)(17,21,22,35)(18,27,24,37)(19,31,32,45)(20,33,23,34)(25,40,29,43)(26,41,28,42)(30,44,36,46)(38,47,39,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,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, 6600215);