#7079064

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1353 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)(19,28,32,40)(20,25,34,44)(21,26,22,37)(24,35,41,31)(29,45,48,39)(30,46,47,38)(33,42,36,43);(9,10)(11,15)(12,14)(13,16)(17,19,21,35,23,32,22,31)(18,24,37,40,27,41,26,28)(20,29,36,30,34,48,33,47)(25,38,42,39,44,46,43,45);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,38)(26,43)(27,44)(28,45)(30,35)(31,47)(32,48)(37,42)(39,40)(41,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,33)(22,36)(23,34)(24,39)(26,42)(27,44)(28,46)(29,35)(31,48)(32,47)(37,43)(38,40)(41,45);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,32,35)(20,33,34,36)(24,40,41,28)(25,42,44,43)(29,30,48,47)(38,45,46,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=g,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=ehi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6986260);