Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,15,9,16)(7,17,20,29,21,12,19,28)(8,22,26,31,24,32,25,13)(18,34,23,27,35,30,36,33);(1,2)(3,4)(5,7,11,21)(6,12,15,17)(8,23,24,36)(9,20,10,19)(13,30,31,34)(14,29,16,28)(18,25,35,26)(22,27,32,33);(2,4)(5,8)(6,13,15,31)(7,18)(9,25)(10,26)(11,24)(12,27,17,33)(14,22,16,32)(19,36)(20,23)(21,35)(28,30,29,34);(6,14)(9,10)(12,28)(13,22)(15,16)(17,29)(19,20)(23,36)(25,26)(27,30)(31,32)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286356, #72287020, #72287028 |
| maximal normal subgroup of | #72286356, #72287020, #72287028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296192);