Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,21,25,36)(9,26)(10,14)(11,15)(12,18)(13,24,30,23)(17,29,22,27)(19,28)(20,32)(31,34,35,33);(5,7)(6,12,15,28)(8,22,25,17)(9,18)(10,19)(11,20)(13,29)(14,16,26,32)(21,35)(23,34,24,33)(27,30)(31,36);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,21,15,30,26,36)(7,17,18,33,20,22,19,34)(12,27,16,31,28,29,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265544, #72265556, #72265570, … |
| maximal normal subgroup of | #72265544, #72265556, #72265570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30952);