#8092027

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

4255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 7999223);