Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,16,14,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(1,3)(2,7)(4,14)(5,11)(6,12)(8,16)(9,15)(10,13)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(7,15)(10,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257614, #72257691, #72257708 |
| maximal normal subgroup of | #72257614, #72257691, #72257708 |
| maximal quotient of | #72264371 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22396);