#7182818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7090014);