Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3894 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)(3,11)(4,8)(5,17)(6,10)(7,13)(9,19)(12,25)(14,24)(15,23)(16,29)(18,20)(21,27)(22,28)(26,32)(30,31);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,24)(10,20)(11,17)(13,19)(15,26)(16,30)(22,32)(23,31)(25,29)(27,28);(4,15)(7,20)(11,24)(12,26)(16,28)(21,31)(25,32)(27,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,21)(9,22)(10,23)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(20,31)(24,32);(5,19)(7,20)(9,17)(13,18)(16,28)(21,31)(22,29)(27,30)Polycyclic presentation:
2^9|f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dhi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233206, #72249649, #72251420, … |
| maximal normal subgroup of | #72233206, #72249649, #72251420, … |
| maximal quotient of | #72233138, #72233417, #72237986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407057);