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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,27,28,24)(20,25,30,32)(21,31,29,26);(2,6)(4,7)(10,13)(14,15)(18,23)(19,28)(20,30)(26,31);(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,24,23,27)(20,29,30,21)(25,31,32,26);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,31)(30,32);(1,4,5,7)(2,3,6,8)(17,20)(18,25)(19,29)(21,28)(22,30)(23,32)(24,31)(26,27);(1,3)(2,7)(4,6)(5,8)(17,21,22,29)(18,26,23,31)(19,20,28,30)(24,25,27,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303876 |
| maximal normal subgroup of | #72303876 |
| maximal quotient of | #72304072, #72304086 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000576);