Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34911 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10,9,11,7,12)(14,18,20,23)(19,25,26,28);(6,10)(7,8)(11,12)(13,14,16,23,17,20,24,18)(15,19,22,28,21,26,27,25);(2,4)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(6,11)(10,12)(15,21)(19,26)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286522 |
| maximal normal subgroup of | #72286522 |
| maximal quotient of | #72286813 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100719);