#8093240

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

4418 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000436);