#8718461

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

4468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,f^2=g,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, 8625657);