#8094670

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

4453 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001866);