Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115377 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,25,26,31)(20,23)(21,24)(22,27,30,28)(29,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(22,30)(25,31);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,22)(20,28)(21,26)(23,31)(24,30)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,g^b=gh,d^c=di,e^c=ei,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233314, #72257408, #72257555, … |
| maximal normal subgroup of | #72233314, #72257408, #72257555, … |
| maximal quotient of | #72233312, #72257797, #72257857 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22574);