#8720803

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

4533 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=i,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, 8627999);