Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10320 |
| 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 | #1439 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,12,22,16)(11,15,20,18,21,23,19,24);(1,2)(3,7,8,5)(4,6)(9,11)(10,15)(12,23)(13,20)(14,21)(16,24)(17,18)(19,22);(1,3)(2,5)(4,8)(6,7)(9,12)(10,14)(11,18)(13,16)(15,19)(17,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fhi,d^c=dh,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279550 |
| maximal normal subgroup of | #72279550 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23509);