#8183728

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

4556 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8090924);