#8720811

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

4545 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:

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

Polycyclic presentation:

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

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, 8628007);