Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
996 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)(5,6,11,17,12,19,29,18)(7,20,25,34,23,32,26,35)(8,14,15,10,27,31,16,9)(13,30,33,24,21,28,36,22);(2,4)(5,7)(6,13)(8,20)(9,28)(10,30)(11,25)(12,23)(14,22)(15,34)(16,35)(17,33)(18,36)(19,21)(24,31)(26,29)(27,32);(5,8)(6,14)(7,21)(9,18)(10,17)(11,15)(12,27)(13,23)(16,29)(19,31)(20,30)(22,35)(24,34)(25,36)(26,33)(28,32)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216647, C2×#147937, #72224879, … |
| maximal normal subgroup of | #72216647, C2×#147937, #72224879, … |
| maximal quotient of | #72216515, #72216518, C2×#147937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55133);