Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198522 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60752 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,17)(3,12,14,32)(4,16)(9,22,11,23)(13,31,18,15)(20,29)(21,30)(25,26);(1,2,4,17,19,9,29,23)(3,8,18,25,31,24,12,30)(5,11,16,22,6,10,20,7)(13,21,15,27,32,26,14,28);(1,3)(2,8)(4,18)(5,15)(6,14)(7,21)(9,24)(10,28)(11,27)(12,29)(13,20)(16,32)(17,25)(19,31)(22,26)(23,30);(2,9)(7,22)(8,24)(10,11)(17,23)(21,26)(25,30)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi68 conjugacy classes — view the full character table.
| maximal quotient of | #72239052 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105718);