Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#476 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,20,33)(8,13)(9,23)(10,15)(11,16)(12,21,28,36)(14,25)(18,34,19,30)(22,32)(24,27,35,29)(26,31);(1,2)(3,4)(7,18,20,19)(12,27,28,29)(17,30,33,34)(21,24,36,35);(2,4)(5,7)(6,12)(8,19)(9,24)(10,18)(11,20)(13,27)(14,30)(15,29)(16,28)(17,31)(21,22)(23,33)(25,36)(26,35)(32,34);(5,8,11,10)(6,13,16,15)(7,19,20,18)(9,22,26,25)(12,27,28,29)(14,23,32,31)(17,30,33,34)(21,35,36,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307420 |
| maximal normal subgroup of | #72307420 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406161);