Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
793 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(22,28)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312175, #72312178, #72312749, … |
| maximal normal subgroup of | #72312175, #72312178, #72312749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23556);