Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1590 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)(5,6,10,14,11,15,9,26)(7,16,19,29,20,28,18,12)(8,21,24,31,25,35,23,13)(17,27,32,30,33,34,22,36);(2,4)(5,7)(6,12,15,29)(8,22,25,32)(9,18)(10,19)(11,20)(13,30)(14,16,26,28)(17,23,33,24)(21,34)(27,35)(31,36);(5,8,9,23,11,25,10,24)(6,13,14,21,15,31,26,35)(7,17,19,32,20,33,18,22)(12,27,28,36,29,34,16,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265200, #72265227, #72265259, … |
| maximal normal subgroup of | #72265200, #72265227, #72265259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31120);