#8359747

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

4549 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)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,24,21,22)(26,32)(28,33)(34,39)(36,40);(3,7)(6,8)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(25,26,31,32)(27,33,29,28)(30,34,38,39)(35,40,37,36);(1,2)(3,6)(4,5)(7,8)(11,18)(14,16)(19,21)(22,24)(25,27,31,29)(26,33,32,28)(30,37,38,35)(34,36,39,40);(1,3,4,7)(2,6,5,8)(9,12,15,10)(11,14,18,16)(13,17,23,20)(19,24,21,22)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,29,31,27)(26,33,32,28)(30,37,38,35)(34,40,39,36);(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,21)(23,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,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, 8266943);