Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,7,12,4,10)(3,14,15,30,17,9,21,28)(5,11,18,29,20,24,22,31)(8,23,26,19,27,32,25,16);(1,3)(2,8)(4,15)(5,16)(6,21)(7,17)(9,24)(10,25)(11,14)(12,27)(13,26)(18,32)(19,20)(22,23)(28,31)(29,30);(1,3)(2,9)(4,15)(5,19)(6,21)(7,17)(8,24)(10,28)(11,27)(12,14)(13,30)(16,20)(18,23)(22,32)(25,31)(26,29);(1,4,7,6)(2,10,12,13)(3,15,17,21)(5,18,20,22)(8,25,27,26)(9,28,14,30)(11,29,24,31)(16,32,19,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261390, #72261398, #72261563, … |
| maximal normal subgroup of | #72261390, #72261398, #72261563, … |
| maximal quotient of | #72303264, #72303570, #72303814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400796);