Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185267 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62824 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,21)(4,15)(8,20,24,32)(9,19,11,16)(13,27)(17,30)(22,28,23,31)(25,29);(1,2)(3,8)(4,16)(5,9)(6,11)(7,17)(10,18)(12,22)(13,28)(14,23)(15,21)(19,30)(20,25)(24,26)(27,32)(29,31);(3,12)(4,15)(7,19)(8,22)(13,29)(14,26)(16,21)(17,30)(20,32)(23,24)(25,27)(28,31);(1,3,5,14)(2,8,9,23)(4,13,15,27)(6,12,18,26)(7,20,19,31)(10,24,11,22)(16,28,21,32)(17,25,30,29)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232880, #72234666, #72234685, … |
| maximal normal subgroup of | #72232880, #72234666, #72234685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92464);