Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464719 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12,18,27)(8,13)(9,16)(10,15)(11,14)(17,33,34,25)(19,28,20,26)(21,29,35,36)(22,32)(23,30)(24,31);(1,2,3,4)(5,7,11,20,9,18,10,19)(6,12,14,26,16,27,15,28)(8,21,24,17,22,35,23,34)(13,29,31,33,32,36,30,25);(2,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,30)(15,31)(16,32)(18,34)(19,35)(20,21)(26,36)(27,33)(28,29);(6,14,16,15)(8,22)(12,26,27,28)(13,30,32,31)(17,34)(21,35)(23,24)(25,36,33,29)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72255619, #72256181, #72256189, … |
| maximal normal subgroup of | #72255619, #72256181, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371917);