Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1701 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)(7,12)(8,19)(9,11)(10,15)(13,22)(14,24)(16,28)(17,27)(18,29)(20,33)(21,31)(23,25)(26,35)(30,34)(32,36);(6,11,15,14)(12,27,29,25)(13,26,19,31)(28,33,36,30);(2,4)(5,7)(6,12)(8,20)(9,23)(10,18)(11,25)(13,30)(14,27)(15,29)(16,21)(17,24)(19,33)(22,34)(26,36)(28,31)(32,35);(5,8,9,21,10,22,24,35)(6,13,14,31,15,19,11,26)(7,16,17,20,18,32,23,34)(12,28,25,30,29,36,27,33)Polycyclic presentation:
2^9|b^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279407 |
| maximal normal subgroup of | #72279407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371436);