Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,17,11)(8,12,19,24)(9,14)(10,13)(16,27)(18,28,31,23)(20,25,32,34)(21,26)(22,29)(30,35)(33,36);(1,2)(3,4)(5,7,9,18,10,17,21,31)(6,11,14,23,13,15,26,28)(8,16,20,30,19,29,32,36)(12,22,25,33,24,27,34,35);(2,4)(5,8)(6,12,13,24)(7,16)(9,20)(10,19)(11,22,15,27)(14,25,26,34)(17,29)(18,30)(21,32)(23,33,28,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30923);