Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,10,6,9,18,24)(3,11,14,25,13,7,17,20)(4,8,16,23,15,22,28,32)(12,19,27,30,26,21,29,31);(1,3)(2,7)(4,12)(5,17)(6,13)(8,21)(9,11)(10,25)(14,18)(15,26)(16,29)(19,22)(20,24)(23,30)(27,28)(31,32);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,23)(11,21)(13,26)(14,27)(17,29)(18,28)(20,30)(24,32)(25,31);(2,9)(5,18)(7,11)(8,22)(14,17)(16,28)(19,21)(27,29);(4,15)(8,22)(12,26)(16,28)(19,21)(23,32)(27,29)(30,31)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,g^b=gi,h^b=hi,e^c=ei,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72249649, #72249652, #72255657, … |
| maximal normal subgroup of | #72249649, #72249652, #72255657, … |
| maximal quotient of | #72240678, #72240682, #72262303 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281391);