#7081653

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(17,18,23,27)(19,29,33,47)(20,25,21,26)(22,37,38,28)(24,39,42,32)(30,46,34,41)(31,43,48,40)(35,45,36,44);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32,23,33,38,39)(18,24,28,29,27,42,37,47)(20,34,35,31,21,30,36,48)(25,43,44,41,26,40,45,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,23)(22,35)(24,40)(26,27)(28,44)(29,46)(31,39)(32,48)(33,34)(36,38)(37,45)(41,47)(42,43);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,20)(18,26,27,25)(19,31,33,48)(22,35,38,36)(24,41,42,46)(28,44,37,45)(29,43,47,40)(30,32,34,39);(10,14)(13,15)(18,27)(20,21)(24,42)(28,37)(29,47)(30,34)(31,48)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,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, 6988849);