Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C23 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,11,16,22)(8,12)(9,14)(10,13)(15,27,28,21)(17,23,30,33)(18,24)(19,25)(20,26)(29,35,36,32)(31,34);(5,7)(6,11,13,22)(8,15)(9,17)(10,16)(12,21,24,27)(14,23,26,33)(18,28)(19,29)(20,30)(25,32,34,35)(31,36);(2,4)(5,8,9,19,10,18,20,31)(6,12,14,25,13,24,26,34)(7,15,17,29,16,28,30,36)(11,21,23,32,22,27,33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,e^b=ei,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237698, #72237703, #72237708, … |
| maximal normal subgroup of | #72237698, #72237703, #72237708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30691);