Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14419 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,21,24,31)(5,10,23,11)(6,13)(7,12)(8,14)(9,18)(16,26,25,27)(17,30,22,29)(19,28)(20,32);(1,3)(2,9)(4,22)(5,25)(6,19)(7,18)(8,20)(10,29)(11,30)(12,32)(13,15)(14,28)(16,23)(17,24)(21,26)(27,31);(1,4,7,23,8,24,6,5)(2,10,12,21,14,11,13,31)(3,16,19,22,20,25,18,17)(9,26,15,29,28,27,32,30);(1,5,6,24,8,23,7,4)(2,11,12,31,14,10,13,21)(3,17,18,25,20,22,19,16)(9,27,15,30,28,26,32,29)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215981, #72219685, #72219692, … |
| maximal normal subgroup of | #72215981, #72219685, #72219692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293246);