Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,9,16,10,15)(7,12,20,28,18,27,19,26)(8,21,23,31,22,30,24,13)(17,32,34,29,33,36,35,25);(1,2)(3,4)(5,7,9,18)(6,12,16,27)(8,17,22,33)(10,20,11,19)(13,29,31,25)(14,26,15,28)(21,32,30,36)(23,35,24,34);(2,4)(5,8,9,22)(6,13)(7,17,18,33)(10,23,11,24)(12,25)(14,30)(15,21)(16,31)(19,34,20,35)(26,32)(27,29)(28,36);(6,14,16,15)(7,18)(12,26,27,28)(13,30,31,21)(17,33)(19,20)(25,32,29,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286134, #72286186, #72286224 |
| maximal normal subgroup of | #72286134, #72286186, #72286224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290931);