Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196266 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,14,9,25)(7,16,19,29,20,12,18,27)(8,21,24,28,22,35,23,13)(17,31,34,26,32,36,33,30);(1,2)(3,4)(5,7,11,20)(6,12,14,16)(8,17,22,32)(9,19,10,18)(13,26,28,30)(15,29,25,27)(21,31,35,36)(23,34,24,33);(2,4)(5,8)(6,13,14,28)(7,17)(9,23)(10,24)(11,22)(12,26,16,30)(15,21,25,35)(18,33)(19,34)(20,32)(27,36,29,31);(6,14)(9,10)(12,16)(13,28)(18,19)(23,24)(26,30)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286845 |
| maximal normal subgroup of | #72286845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281243);