Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,11,16,25)(8,18)(9,21)(10,13)(12,20)(14,22)(15,28,30,33)(17,32,31,24)(19,34)(23,36,26,29)(27,35);(1,2)(3,4)(5,7)(6,11,13,25)(8,15)(9,17)(10,16)(12,26,27,23)(14,24,21,32)(18,33,34,28)(19,30)(20,29)(22,31)(35,36);(2,4)(5,8,9,20,10,19,22,35)(6,12,14,18,13,27,21,34)(7,15,17,29,16,30,31,36)(11,23,24,28,25,26,32,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237703, #72237704, #72237714, … |
| maximal normal subgroup of | #72237703, #72237704, #72237714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31100);