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 | |
5274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,16)(18,19)(23,25)(24,26)(30,31);(9,10,12,13)(11,14,15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(2,6)(3,7)(18,19)(20,27)(24,26)(28,32);(1,3)(2,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,4)(2,3)(5,8)(6,7)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318230, #72318231, #72318247, … |
| maximal normal subgroup of | #72318230, #72318231, #72318247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646661);