Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | #23386 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1427 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)(8,19)(21,32)(24,35)(29,36);(2,4)(5,6)(7,15)(8,12)(9,22)(10,14)(11,17)(13,20)(16,29)(18,28)(19,26)(21,33)(23,25)(24,31)(27,32)(30,36)(34,35);(5,7)(6,11)(8,16)(9,23)(10,18)(12,24)(13,28)(14,25)(15,22)(17,20)(19,30)(21,34)(26,35)(27,36)(29,33)(31,32);(5,8,20,32,10,19,9,21)(6,12,13,27,14,26,22,33)(7,16,17,31,18,30,23,34)(11,24,28,36,25,35,15,29)Polycyclic presentation:
2^9|d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257124, #72257207, #72257214 |
| maximal normal subgroup of | #72257124, #72257207, #72257214 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321096);