Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1598 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)(5,6,9,14,10,13,22,28)(7,15,18,30,17,11,23,25)(8,12,21,26,20,27,33,35)(16,24,31,34,19,29,32,36);(2,4)(5,7)(6,11,13,15)(8,19)(9,18)(10,17)(12,24,27,29)(14,25,28,30)(16,20)(21,32)(22,23)(26,34,35,36)(31,33);(5,8)(6,12)(7,16,17,19)(9,21)(10,20)(11,24,15,29)(13,27)(14,26)(18,31,23,32)(22,33)(25,34,30,36)(28,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237652, #72237655, #72237664, … |
| maximal normal subgroup of | #72237652, #72237655, #72237664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31230);