#7081521

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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)(19,28,33,45)(20,25,34,24)(21,36,29,26)(23,38,41,32)(30,46,47,40)(31,39,48,43)(35,42,37,44);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,29,38,22,33,21,32)(18,23,36,45,27,41,26,28)(20,30,37,31,34,47,35,48)(24,39,44,46,25,43,42,40);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,30)(21,35)(22,34)(23,39)(25,27)(26,42)(28,40)(29,37)(31,38)(32,48)(33,47)(36,44)(41,43)(45,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,31)(21,37)(22,34)(23,40)(24,27)(26,42)(28,39)(29,35)(30,38)(32,47)(33,48)(36,44)(41,46)(43,45);(10,14)(13,15)(20,34)(24,25)(30,47)(31,48)(35,37)(39,43)(40,46)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,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=eh,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, 6988717);