#8094585

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

4473 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)(10,16)(11,14)(12,22)(21,24)(28,36)(32,37)(34,40)(38,39);(2,5)(3,7)(9,10)(11,19)(12,13)(14,17)(15,16)(18,22)(20,21)(23,24)(26,30)(31,33)(32,37)(38,39);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,17,16,19)(12,20,22,23)(13,24,18,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,20)(13,16)(14,23)(15,22)(17,21)(19,24);(9,13,15,18)(10,12,16,22)(11,21,14,24)(17,23,19,20)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,14,15,11)(10,17,16,19)(12,23,22,20)(13,24,18,21)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=ghi,f^2=g,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, 8001781);