Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13584 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| socle | C2 | C22×#353 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1131 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)(3,12,15,7)(4,8)(5,10,19,9)(6,11)(13,22,18,20)(14,26,28,21)(16,24,30,23)(17,25)(27,32,29,31);(1,3)(2,7)(4,14)(5,18)(6,13)(8,21)(9,22)(10,12)(11,20)(15,19)(16,29)(17,27)(23,32)(24,26)(25,31)(28,30);(3,13)(7,20)(12,22)(14,27)(15,18)(21,31)(26,32)(28,29);(1,4,6,17)(2,8,11,25)(3,14,13,27)(5,16,19,30)(7,21,20,31)(9,23,10,24)(12,26,22,32)(15,28,18,29)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^b=ehi,f^b=fgh,g^b=gi,f^e=fhi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279376, #72279391, #72279413, … |
| maximal normal subgroup of | #72279376, #72279391, #72279413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419797);