#8429298

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

4309 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336494);