Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,12,8,11,23)(7,16,20,32,21,33,19,31)(9,24,26,22,28,36,27,14)(13,25,30,35,17,34,29,18);(1,2)(3,4)(5,7)(6,13)(8,17)(9,25)(10,20)(11,19)(12,21)(14,31)(15,30)(16,24)(18,27)(22,32)(23,29)(26,35)(28,34)(33,36);(2,4)(5,8)(6,12)(7,17)(9,14)(10,15)(11,23)(13,21)(16,18)(19,29)(20,30)(22,28)(24,27)(25,31)(26,36)(32,34)(33,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250059, #72250071, #72250075, … |
| maximal normal subgroup of | #72250059, #72250071, #72250075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54862);