Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1076 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)(7,17,20,31)(8,22,24,14)(9,25,26,13)(10,27)(11,16)(12,18)(15,23)(19,35)(21,28,33,32)(29,36)(30,34);(2,4)(5,7,10,21,11,20,23,33)(6,12,15,29,16,30,27,35)(8,18,9,19,24,34,26,36)(13,28,14,17,25,32,22,31);(6,13)(12,28)(14,15)(16,25)(17,29)(22,27)(30,32)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,f^b=fhi,f^c=fh,f^d=fi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308038, #72312394, #72312398, … |
| maximal normal subgroup of | #72308038, #72312394, #72312398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53865);