Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #37674 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(18,24)(19,20)(21,29)(31,32);(2,5)(6,8)(10,13)(14,15)(17,18,23,24)(19,22,20,27)(21,30,29,25)(26,32,28,31);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,22,24,27)(21,28,29,26)(25,32,30,31);(3,7)(6,8)(9,11,12,16)(10,14,13,15)(17,20,23,19)(18,22,24,27)(21,28,29,26)(25,31,30,32);(1,2)(3,6)(4,5)(7,8)(17,21,23,29)(18,25,24,30)(19,26,20,28)(22,31,27,32);(1,3)(2,6)(4,7)(5,8)(17,22)(18,20)(19,24)(21,31)(23,27)(25,28)(26,30)(29,32)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506739);