#8718530

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

4670 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gi,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, 8625726);