Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(20,28)(23,26)(24,30)(27,31);(10,13)(14,16)(17,18,20,22,21,24,28,30)(19,26,25,32,27,23,31,29);(9,11,12,15)(10,14,13,16)(17,19,20,25,21,27,28,31)(18,23,22,29,24,26,30,32);(2,4)(18,24)(22,30)(23,26)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240744, #72240778, #72240780, … |
| maximal normal subgroup of | #72240744, #72240778, #72240780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299517);