#8359744

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

4400 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)(9,10,15,13)(11,12,19,16)(14,17,22,21)(18,20,24,23)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(3,7)(6,8)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,2)(3,6)(4,5)(7,8)(14,22)(17,21)(18,24)(20,23)(27,34)(30,32)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=hi,e^2=g,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, 8266940);