#7149416

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

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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, 7056612);