#6687955

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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

1532 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)(9,10,12,14)(11,15,16,13)(19,29)(24,38)(30,42)(31,37)(32,43)(39,47)(44,48)(45,46);(1,3)(2,5)(4,8)(6,7)(17,18,22,27)(19,30,29,42)(20,28,23,25)(21,34,36,40)(24,37,38,31)(26,35,41,33)(32,39,43,47)(44,45,48,46);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,31)(21,32)(22,29)(23,37)(25,30)(26,39)(27,38)(28,42)(33,45)(34,44)(35,46)(36,43)(40,48)(41,47);(2,6)(5,7)(17,20,22,23)(18,25,27,28)(19,31,29,37)(21,35,36,33)(24,30,38,42)(26,40,41,34)(32,46,43,45)(39,48,47,44);(3,8)(5,7)(17,21,22,36)(18,26,27,41)(19,32,29,43)(20,33,23,35)(24,39,38,47)(25,34,28,40)(30,44,42,48)(31,45,37,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6595151);