Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | #23798 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22,27,31,26,29,25,35)(23,32,24,30,33,36,34,28);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(21,23,27,24,26,33,25,34)(22,28,35,36,29,30,31,32);(1,2,3,4)(5,7,9,14)(6,12,13,19)(8,15,11,18)(10,16,17,20)(22,29)(23,33)(24,34)(31,35);(7,14)(12,19)(15,18)(16,20)(21,24,27,33,26,34,25,23)(22,30,31,36,29,28,35,32)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=fh,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=di,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120201);