#7081526

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=gi,e^2=i,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, 6988722);