#6694457

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=gi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6601653);