#8359134

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

4499 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8266330);