Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(19,24)(21,23);(1,2,3,4)(17,18)(19,21)(20,22)(23,24);(10,13)(11,16)(14,15)(18,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fi,e^c=eh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217667 |
| maximal normal subgroup of | #72217667 |
| maximal quotient of | #72216703, #72216704, #72216706 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23021);