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 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1749 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)(7,17)(8,13)(9,14)(10,26)(11,16)(12,20)(15,24)(18,28)(19,27)(21,32)(22,35)(23,31)(25,30)(29,34)(33,36);(5,7)(6,12,16,29)(8,22)(9,19)(10,20)(11,21)(13,27,31,36)(14,28,30,35)(15,17,26,32)(18,23)(24,34)(25,33);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,23)(12,27)(14,15)(16,31)(17,28)(19,20)(21,22)(26,30)(29,36)(32,35)(33,34);(5,9,24,8,11,25,10,23)(6,14,26,31,16,30,15,13)(7,19,34,22,21,33,20,18)(12,28,32,36,29,35,17,27)Polycyclic presentation:
2^9|b^2=gi,d^2=ei,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 | #72250742, #72250753, #72250788, … |
| maximal normal subgroup of | #72250742, #72250753, #72250788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371439);