Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21379 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,12,19,27,7,3,8,14,20,10,21,25)(4,13,16,26,17,22,29,31,15,9,18,23,28,24,30,32);(1,3,5,14,6,10,19,25,7,2,8,11,20,12,21,27)(4,13,18,23,17,22,30,32,15,9,16,26,28,24,29,31);(1,4)(2,9)(3,13)(5,18)(6,17)(7,15)(8,16)(10,22)(11,26)(12,24)(14,23)(19,30)(20,28)(21,29)(25,32)(27,31);(2,10,3,12)(5,8)(9,22,13,24)(11,27,14,25)(16,18)(19,21)(23,32,26,31)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231607, #72231615, … |
| maximal normal subgroup of | #72213937, #72231607, #72231615, … |
| maximal quotient of | #72231501, #72237207, C2×#495505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402701);