#8702846

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^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, 8610042);