Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363904 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13956 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2539 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)(3,12,15,27)(4,16,18,8)(5,10)(6,11)(7,14,22,30)(9,17,24,31)(13,26)(19,25)(20,32)(21,29)(23,28);(1,3)(2,7,11,22)(4,13)(5,14)(6,15)(8,23,16,20)(9,21,24,26)(10,12,25,27)(17,28)(18,29)(19,30)(31,32);(1,4)(2,8)(3,13)(5,17)(6,18)(7,20)(9,10)(11,16)(12,26)(14,28)(15,29)(19,31)(21,27)(22,23)(24,25)(30,32);(1,4)(2,9,11,24)(3,13)(5,17)(6,18)(7,21,22,26)(8,10,16,25)(12,20,27,23)(14,28)(15,29)(19,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261408, #72261457, #72319325, … |
| maximal normal subgroup of | #72261408, #72261457, #72319325, … |
| maximal quotient of | #72211251 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271100);