Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2246 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)(7,8)(13,14,17,21,18,16,20,25)(15,19,24,27,22,23,26,28);(2,4)(9,10,11,12)(13,15)(14,19)(16,23)(17,24)(18,22)(20,26)(21,27)(25,28);(6,8)(15,22)(19,23)(24,26)(27,28);(13,16,17,25,18,14,20,21)(15,23,24,28,22,19,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,g^2=i,b^a=be,c^a=cf,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286847, #72317350, #72317384, … |
| maximal normal subgroup of | #72286847, #72317350, #72317384, … |
| maximal quotient of | C2×#354295, #72321451 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261491);