#8095913

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

4400 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003109);