#6692655

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(19,25)(20,26)(21,36)(22,28)(23,27)(24,29)(30,40)(31,42)(32,43)(33,44)(34,46)(35,47)(37,48)(38,41)(39,45);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,24,32)(18,25,29,43)(20,33,35,37)(21,31,23,38)(22,34,30,39)(26,44,47,48)(27,41,36,42)(28,46,40,45);(9,12,13,11)(10,14,15,16)(17,20)(18,26)(19,30)(21,34)(22,32)(23,39)(24,35)(25,40)(27,45)(28,43)(29,47)(31,37)(33,38)(36,46)(41,44)(42,48);(2,6)(5,7)(17,21,24,23)(18,27,29,36)(19,31,32,38)(20,34,35,39)(22,33,30,37)(25,41,43,42)(26,45,47,46)(28,48,40,44);(3,8)(5,7)(17,22,24,30)(18,28,29,40)(19,20,32,35)(21,37,23,33)(25,26,43,47)(27,44,36,48)(31,39,38,34)(41,46,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 6599851);