Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1420 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)(3,7,8,5)(4,6)(9,10,13,18,15,11,14,17)(12,20,22,19,23,24,21,16);(9,11)(10,15)(12,19)(13,17)(14,18)(16,23)(20,21)(22,24);(1,3)(2,5)(4,8)(6,7)(9,12,13,21,15,23,14,22)(10,16,17,20,11,19,18,24)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^c=fh,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72267251 |
| maximal normal subgroup of | #72267251 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26654);