Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,18,11)(8,12)(9,22)(10,13)(14,24)(16,31,33,25)(17,26,35,32)(19,27,34,36)(20,29)(21,30)(23,28);(5,7)(6,11)(8,19,20,34)(9,17)(10,18)(12,27,29,36)(13,15)(14,26)(16,21,33,23)(22,32)(24,35)(25,28,31,30);(1,2,3,4)(5,8)(6,12)(7,16)(9,23)(10,20)(11,25)(13,29)(14,30)(15,31)(17,34)(18,33)(19,35)(21,24)(22,28)(26,36)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,g^b=gi,d^c=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280130, #72280335, #72280467, … |
| maximal normal subgroup of | #72280130, #72280335, #72280467, … |
| maximal quotient of | #72303279 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31158);