#8630862

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

4590 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)(3,4)(5,8)(6,7)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,3)(2,4)(5,6)(7,8)(10,15)(11,19)(12,22)(21,23)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(1,4)(2,5)(3,7)(6,8)(10,15)(12,22)(16,18)(20,24)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(1,3,6,5)(2,4,7,8)(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,23,17,21)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,5,6,3)(2,4,7,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538058);