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 | #1762 |
| 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)(7,18)(8,15)(9,25)(10,26)(11,16)(12,28)(13,17)(14,19)(20,32)(21,29)(22,35)(23,30)(24,33)(27,34)(31,36);(5,7,10,20,11,27,8,23,13,22,9,19,12,21,24,36)(6,14,15,18,28,30,26,29,17,32,33,35,16,31,25,34);(5,8,11,9,13,24,12,10)(6,15,16,25,17,33,28,26)(7,19,21,23,22,20,27,36)(14,29,30,35,32,34,31,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=hi,g^2=i,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 | #72250625, #72250639, #72250653, … |
| maximal normal subgroup of | #72250625, #72250639, #72250653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53543);