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 | |
6310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(18,23)(19,25)(20,27)(28,32);(2,6)(3,8)(11,15)(14,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32);(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)Polycyclic presentation:
2^9|b^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72326449 |
| maximal normal subgroup of | #72326449 |
| maximal quotient of | #72326466, #72326488, #72326704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8340871);