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 | |
6118 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,29)(23,25)(26,30)(27,31)(28,32);(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,24,25,22)(21,27,30,28)(26,32,29,31);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(18,25)(19,20)(21,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,31,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,32)(25,29);(17,22)(18,19)(20,25)(21,31)(23,24)(26,27)(28,29)(30,32)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72249465, #72249491, #72249540, … |
| maximal normal subgroup of | #72249465, #72249491, #72249540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617674);