#8127402

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

4486 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,20)(15,18)(17,21)(19,22)(23,24)(26,30)(28,35)(31,37)(33,36);(1,2,4,6)(3,5,8,7)(11,18)(15,16)(19,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,18)(10,15,12,16)(13,19,21,24)(17,22,20,23)(25,27,29,34)(26,31,30,37)(28,36,35,33)(32,40,38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,28,29,35)(26,32,30,38)(27,33,34,36)(31,39,37,40);(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,27)(26,31)(28,33)(29,34)(30,37)(32,39)(35,36)(38,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034598);