Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,10)(8,16)(11,17)(12,18)(13,20)(15,19)(21,22,25,29,26,30,34,28)(23,31,24,27,32,35,33,36);(5,6,8,10,9,12,16,18)(7,13,15,20,14,11,19,17)(21,23,25,24,26,32,34,33)(22,27,28,31,30,36,29,35);(2,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,13)(16,19)(18,20);(6,12)(7,14)(10,18)(15,19)(21,24,25,32,26,33,34,23)(22,27,29,35,30,36,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=dh,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224967, #72285149, C2×#210977, … |
| maximal normal subgroup of | #72224967, #72285149, C2×#210977, … |
| maximal quotient of | #72224959, #72224960, C2×#210977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118173);