Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1046 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)(5,6,9,16,11,17,10,28)(7,18,21,34,22,36,29,35)(8,23,26,15,27,13,25,14)(12,30,31,20,32,24,33,19);(2,4)(5,7)(6,12,17,32)(8,24)(9,21)(10,29)(11,22)(13,18,23,36)(14,34,15,35)(16,31,28,33)(19,26)(20,25)(27,30);(6,13)(8,25)(9,10)(12,18)(14,28)(15,16)(17,23)(19,30)(20,24)(21,29)(26,27)(31,35)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231513, #72231523, #72231541, … |
| maximal normal subgroup of | #72231513, #72231523, #72231541, … |
| maximal quotient of | #72265658, #72279477, C2×#146792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53988);