Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361803 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13903 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,5,20)(3,13,17,25,16,8,22,26)(4,9,18,29,19,10,21,28)(14,23,32,27,31,30,15,24);(1,3,5,17,7,16,6,22)(2,8,20,26,11,13,12,25)(4,15,18,31,19,32,21,14)(9,27,29,23,10,24,28,30);(1,4)(2,9)(3,14)(5,21)(6,18)(7,19)(8,23)(10,11)(12,29)(13,30)(15,22)(16,31)(17,32)(20,28)(24,25)(26,27);(1,4)(2,10)(3,15)(5,18)(6,21)(7,19)(8,24)(9,11)(12,29)(13,27)(14,22)(16,32)(17,31)(20,28)(23,25)(26,30)Polycyclic presentation:
2^9|a^2=h,b^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72338600, #72338606, #72338612 |
| maximal normal subgroup of | #72338600, #72338606, #72338612 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269001);