Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,11,16,10,6,12,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(1,3)(2,6)(4,12)(5,10)(7,9)(8,16)(11,13)(14,15)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(6,9)(14,16)(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,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257624, #72257688, #72257708 |
| maximal normal subgroup of | #72257624, #72257688, #72257708 |
| maximal quotient of | #72264452 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22390);