Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,9)(4,11)(7,14)(8,13)(10,15)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,2,4,6,5,8,11,13)(3,7,9,12,10,14,15,16)(17,19)(18,22,24,29)(20,25)(21,26)(23,28,30,32)(27,31);(1,3,4,9,5,10,11,15)(2,7,6,12,8,14,13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(3,10)(7,14)(9,15)(12,16)(20,27)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112957);