#8527509

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434705);