Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #390 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10)(11,15,20,29)(12,22)(13,25)(14,17)(16,23)(18,26)(19,31,32,40)(21,35,34,28)(24,36)(27,38,39,33)(30,37);(2,4)(5,6)(7,8)(9,11)(10,15,17,29)(12,19)(13,21)(14,20)(16,27,30,39)(18,28,25,35)(22,31,36,40)(23,33)(24,32)(26,34)(37,38);(6,8)(9,12,13,23,14,24,26,37)(10,16,18,22,17,30,25,36)(11,19,21,33,20,32,34,38)(15,27,28,31,29,39,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314513, #72314517, #72314529, … |
| maximal normal subgroup of | #72314513, #72314517, #72314529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31084);