Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62819 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,12,17,27)(4,6,10,18)(5,20,24,9)(8,21,26,14)(13,30,25,31)(15,28,16,29)(19,32,23,22);(1,3)(2,8)(4,13)(5,21)(6,15)(7,17)(9,27)(10,25)(11,26)(12,20)(14,24)(16,18)(19,28)(22,31)(23,29)(30,32);(5,22)(9,23)(12,28)(14,30)(19,20)(21,31)(24,32)(27,29);(1,4,18,2)(3,13,16,8)(5,23,32,20)(6,11,7,10)(9,24,19,22)(12,21,29,30)(14,28,31,27)(15,26,17,25)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213838, #72232843, #72234677, … |
| maximal normal subgroup of | #72213838, #72232843, #72234677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253660);