#8096029

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

4532 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 8003225);