#8720792

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

4420 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,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, 8627988);