#8743429

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

4621 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)(11,20)(12,17)(13,24)(14,19)(15,22)(16,23)(18,21)(26,31)(27,33)(28,36)(37,38);(3,7)(6,8)(9,11)(10,15)(12,13)(14,22)(16,24)(17,18)(19,20)(21,23)(26,31)(27,33)(32,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,23)(10,16,19,17)(11,13,22,21)(15,24,20,18)(25,26,30,31)(27,34,33,29)(28,32,36,39)(35,37,40,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,22)(14,23)(15,24)(16,19)(18,20)(25,27,30,33)(26,29,31,34)(28,35,36,40)(32,38,39,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,18,19,24)(11,12,22,23)(15,17,20,16)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(9,10)(11,15)(12,16)(13,24)(14,19)(17,23)(18,21)(20,22)(25,29)(26,27)(28,37)(30,34)(31,33)(32,40)(35,39)(36,38)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8650625);