Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14438 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11)(4,16)(5,9,19,25)(7,13)(8,18)(12,26,29,32)(14,21)(15,24)(17,30)(20,28,22,27)(23,31);(1,3)(2,7)(4,12)(5,13)(6,15)(8,22)(9,24)(10,21)(11,25)(14,19)(16,20)(17,27)(18,29)(23,26)(28,31)(30,32);(1,4,5,17,6,18,19,31)(2,8,9,23,10,16,25,30)(3,12,13,27,15,29,14,28)(7,20,24,32,21,22,11,26);(2,9,10,25)(3,13,15,14)(7,21)(8,23,16,30)(11,24)(12,27,29,28)(20,22)(26,32)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318883, #72318910, #72318981, … |
| maximal normal subgroup of | #72318883, #72318910, #72318981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293927);