#8073127

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7980323);