Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,14,18,24)(12,17)(13,16)(15,19,23,20)(21,22);(1,3,4,8,5,2,6,7)(9,11)(10,14,16,24)(12,20)(13,18)(15,22,23,17)(19,21);(9,11)(10,15)(12,20)(13,18)(14,22)(16,23)(17,24)(19,21);(2,3)(7,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308938, C2×#196200 |
| maximal normal subgroup of | #72308938, C2×#196200 |
| maximal quotient of | C2×#196200, #72309098, #72309101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103396);