Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2542 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)(7,15,18,11)(8,19,20,29)(9,14)(10,13)(12,21,27,22)(16,31)(17,30,35,25)(23,28)(24,33)(26,34)(32,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,26)(13,15)(14,25)(19,31)(20,32)(21,33)(22,34)(23,35)(24,27)(28,30)(29,36);(5,8)(6,12)(7,16)(9,22)(10,20)(11,24)(13,27)(14,29)(15,26)(17,34)(18,32)(19,28)(21,23)(25,31)(30,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,g^b=gi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237509, #72237516, #72237648, … |
| maximal normal subgroup of | #72237509, #72237516, #72237648, … |
| maximal quotient of | C2×#123554 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30750);