#8425343

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

4689 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,f^2=g,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, 8332539);