Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1514 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C2 | C2×#1514 |
| prime radical | #60693 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
726 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)(3,13,16,30)(4,10)(5,19,21,26)(6,9)(7,12)(8,15)(11,18,28,22)(14,29,17,27)(20,24)(23,32)(25,31);(1,3)(2,8)(4,14)(5,20)(6,17)(7,16)(9,23)(10,24)(11,27)(12,25)(13,19)(15,22)(18,31)(21,32)(26,30)(28,29);(2,9,12,10)(5,21)(8,23,25,24)(11,19,28,26)(13,29,30,27)(15,31)(18,22)(20,32);(1,4,7,6)(2,10,12,9)(3,14,16,17)(5,18,21,22)(8,24,25,23)(11,19,28,26)(13,29,30,27)(15,20,31,32)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ch,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286542, #72286551, #72286573, … |
| maximal normal subgroup of | #72286542, #72286551, #72286573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389142);