Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,18)(9,21)(10,13)(12,20)(14,23)(15,29)(16,24)(17,33)(19,28)(22,27)(25,31)(26,32)(30,36)(34,35);(11,24)(15,30)(26,33)(31,34);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,25)(13,24)(14,26)(18,29)(19,30)(20,31)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8)(6,12)(7,15)(9,22)(10,19)(11,25)(13,27)(14,28)(16,30)(17,34)(18,21)(20,23)(24,35)(26,36)(29,33)(31,32)Polycyclic presentation:
2^9|h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252890, #72252967, #72252977, … |
| maximal normal subgroup of | #72252890, #72252967, #72252977, … |
| maximal quotient of | #72314529, #72314530 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324811);