Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1411 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)(5,6,10,15,11,13,9,23)(7,16,18,12,19,27,24,26)(8,14,22,30,21,28,20,29)(17,31,32,25,33,35,34,36);(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,26)(14,25)(15,27)(16,23)(20,34)(21,33)(22,32)(28,36)(29,31)(30,35);(2,4)(6,13)(8,20)(9,10)(14,29)(16,27)(17,32)(18,24)(21,22)(25,31)(28,30)(33,34)(35,36);(5,8,10,22,11,21,9,20)(6,14,15,30,13,28,23,29)(7,17,18,32,19,33,24,34)(12,25,27,35,26,36,16,31)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226669, #72226676, #72226677 |
| maximal normal subgroup of | #72226669, #72226676, #72226677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288674);