Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136733 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
982 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)(6,12,15,25)(8,9)(10,26)(13,30,32,17)(14,16,22,28)(18,23)(19,35)(20,29)(24,27)(31,34,36,33);(5,6)(7,17)(8,22)(9,25)(10,28)(11,15)(12,27)(13,19)(14,24)(16,26)(18,34)(20,36)(21,30)(23,32)(29,31)(33,35);(2,4)(5,7)(6,13)(8,23)(9,18)(10,29)(11,21)(12,30)(14,31)(15,32)(16,34)(17,25)(19,24)(20,26)(22,36)(27,35)(28,33)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dh,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72224933, #72226807, … |
| maximal normal subgroup of | #72224903, #72224933, #72226807, … |
| maximal quotient of | #72224898, #72237262, C2×#147917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55113);