#7081619

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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)(17,18,22,25)(19,27,31,38)(20,24,33,41)(21,35,36,26)(23,29,39,30)(28,43,32,47)(34,48,44,42)(37,46,40,45);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,30,22,31,36,29)(18,23,26,27,25,39,35,38)(20,32,34,46,33,28,44,45)(24,40,42,47,41,37,48,43);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,41)(26,42)(27,43)(29,46)(30,45)(31,32)(35,48)(36,44)(38,47)(39,40);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,29)(20,33)(21,36)(23,38)(24,41)(26,35)(27,39)(28,45)(30,31)(32,46)(37,43)(40,47);(10,14)(13,15)(18,25)(20,33)(23,39)(26,35)(27,38)(28,32)(34,44)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=g,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, 6988815);