Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481601 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #35050 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1443 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)(3,12,15,22)(4,16,17,26)(5,18,20,9)(6,11)(7,10)(8,13,24,30)(14,28,31,32)(19,23,29,25)(21,27);(1,3)(2,8)(4,13)(5,19)(6,15)(7,14)(9,22)(10,23)(11,25)(12,26)(16,28)(17,29)(18,32)(20,30)(21,31)(24,27);(1,4,6,5)(2,9,11,26)(3,13,15,19)(7,17,21,20)(8,22,25,12)(10,18,27,16)(14,29,31,30)(23,32,24,28);(3,14)(8,23)(12,28)(13,29)(15,31)(19,30)(22,32)(24,25)Polycyclic presentation:
2^9|a^2=f,c^2=h,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285245 |
| maximal normal subgroup of | #72285245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389353);