Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3812 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)(5,6,10,14,11,16,9,15)(7,12,19,27,20,28,18,32)(8,21,23,13,24,30,25,31)(17,29,33,26,34,36,22,35);(2,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,27)(15,32)(16,28)(17,23)(21,35)(24,33)(25,34)(26,30)(31,36);(5,8)(6,13)(7,17)(9,25)(10,23)(11,24)(12,26)(14,30)(15,21)(16,31)(18,22)(19,33)(20,34)(27,36)(28,35)(29,32);(2,4)Polycyclic presentation:
2^9|a^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72322924 |
| maximal normal subgroup of | #72322924 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266682);