Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
670 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)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20)(21,22,25,29,26,30,24,28)(23,31,32,27,33,35,34,36);(5,7)(6,11)(8,15)(9,17)(10,16)(12,18)(13,19)(14,20)(21,23,25,32,26,33,24,34)(22,27,29,35,30,36,28,31);(1,2,3,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12)(21,24,26,25)(22,28,30,29)(23,32,33,34)(27,35,36,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=eh,d^c=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110900);