#6714754

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6621950);