Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2510 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)(7,15)(8,19,22,12)(9,23)(10,14)(11,17)(13,25)(16,32,20,30)(18,31)(21,28,24,29)(26,34,36,33)(27,35);(5,7)(6,11,14,27)(8,20,22,16)(9,17)(10,18)(12,26)(13,15,23,31)(19,36)(21,34,24,33)(25,35)(28,32)(29,30);(2,4)(5,8)(6,12)(7,16)(9,24)(10,22)(11,26)(13,29)(14,19)(15,30)(17,34)(18,20)(21,25)(23,28)(27,36)(31,32)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265200, #72265248, #72265271, … |
| maximal normal subgroup of | #72265200, #72265248, #72265271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30767);