Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C24 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
614 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)(5,6,9,15,12,17,11,26)(7,18,19,30,22,34,23,36)(8,16,25,32,24,33,10,14)(13,27,29,35,31,21,28,20);(2,4)(5,7,10,21,11,23,24,35,12,22,25,27,9,19,8,20)(6,13,16,30,15,29,32,34,17,31,33,36,26,28,14,18);(7,19,22,23)(8,24)(10,25)(13,28,31,29)(14,32)(16,33)(18,30,34,36)(20,21,35,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi,c^b=ch,g^b=gi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239809, #72239831, … |
| maximal normal subgroup of | #72239802, #72239809, #72239831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53541);