Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,8)(9,11)(10,12)(13,14,19,23,17,20,18,26)(15,22,16,21,24,28,25,27);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(21,27)(22,28)(23,26);(1,2,3,4)(5,8,9,10)(6,7,11,12)(13,15,17,24)(14,21,20,27)(16,18,25,19)(22,23,28,26);(6,11)(7,12)(13,16)(14,22)(15,18)(17,25)(19,24)(20,28)(21,23)(26,27)Polycyclic presentation:
2^9|a^2=h,c^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237222, #72264630, #72264635, … |
| maximal normal subgroup of | #72237222, #72264630, #72264635, … |
| maximal quotient of | #72239111 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293089);