#7081657

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,31,45)(20,25,33,24)(21,35,37,26)(23,38,41,32)(29,46,48,40)(30,39,47,43)(34,42,36,44);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,32,22,31,37,38)(18,23,26,28,27,41,35,45)(20,29,34,47,33,48,36,30)(24,39,42,40,25,43,44,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,29)(21,34)(22,33)(23,39)(25,27)(26,42)(28,40)(30,38)(31,48)(32,47)(35,44)(36,37)(41,43)(45,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,33)(23,40)(24,27)(26,42)(28,39)(29,38)(31,47)(32,48)(34,37)(35,44)(41,46)(43,45);(10,14)(13,15)(19,31)(20,33)(23,41)(24,25)(28,45)(32,38)(34,36)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=h,d^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988853);