Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7,6,10,4,9)(3,11,13,8,14,15,12,16)(17,18,23,26)(19,22,25,21)(20,27)(24,28)(29,31)(30,32);(2,7,10,9)(8,15,16,11)(17,19,21,18,23,25,22,26)(20,27,29,24,30,32,28,31);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4,6,5)(2,9,10,7)(3,12,14,13)(8,11,16,15)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,f^b=fi,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72286759, #72286775, #72286796 |
| maximal normal subgroup of | #72286759, #72286775, #72286796 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109333);