#8127350

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

4598 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=gh,e^2=g,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, 8034546);