Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
765 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)(5,6)(7,13)(8,17)(9,11)(10,16)(12,14)(15,19)(18,20)(21,22,27,31,26,29,25,35)(23,32,24,30,33,36,34,28);(5,7,14,20,9,15,8,16)(6,10,17,19,11,18,12,13)(21,23,27,24,26,33,25,34)(22,28,35,36,29,30,31,32);(1,2,3,4)(6,11)(10,18)(12,17)(13,19)(22,29)(23,33)(24,34)(31,35);(7,15)(10,18)(13,19)(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=dh,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120214);