Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,16)(8,10)(9,22)(13,28)(14,23)(15,20)(17,30)(18,33)(19,24)(21,25)(26,34)(29,32)(31,36);(1,2)(3,4)(5,6)(7,17)(8,20)(9,23)(10,12)(11,15)(13,19)(14,25)(16,32)(18,31)(21,27)(22,35)(24,30)(26,29)(28,33)(34,36);(2,4)(5,7,8,19,11,18,10,26)(6,13,12,17,15,29,20,31)(9,24,21,16,25,34,22,33)(14,28,27,36,23,32,35,30);(7,18)(13,29)(16,33)(17,31)(19,26)(24,34)(28,32)(30,36)Polycyclic presentation:
2^9|c^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250068, #72250075, #72250107, … |
| maximal normal subgroup of | #72250068, #72250075, #72250107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406229);