Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,12)(9,22)(10,15)(11,14)(13,18)(16,23)(19,28)(20,33)(21,31)(24,30)(25,32)(26,34)(27,35)(29,36);(6,12)(7,18)(8,10)(9,23)(13,28)(14,15)(16,31)(17,27)(19,24)(20,36)(21,25)(26,35)(29,34)(30,33);(2,4)(5,7,8,19,11,20,10,26)(6,13,15,30,14,29,12,27)(9,24,21,18,25,35,23,36)(16,28,32,17,31,34,22,33);(6,14)(12,15)(13,29)(16,31)(17,33)(22,32)(27,30)(28,34)Polycyclic presentation:
2^9|c^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gh,h^b=hi,f^c=fhi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231495, #72231532, #72231547, … |
| maximal normal subgroup of | #72231495, #72231532, #72231547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406275);