Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,12,7,11,18)(8,19,22,16,23,29,21,13)(9,14,25,31,27,17,26,32)(20,34,35,33,36,30,24,28);(1,2)(3,4)(5,7)(6,12)(8,19,23,29)(9,17)(10,15)(11,18)(13,21,16,22)(14,27)(20,30,36,34)(24,28,35,33)(25,31)(26,32);(2,4)(5,8,10,21,12,23,11,22)(6,13,15,29,7,16,18,19)(9,24,25,36,27,35,26,20)(14,30,31,33,17,34,32,28);(5,9,12,27)(6,14,7,17)(8,20,23,36)(10,26,11,25)(13,28,16,33)(15,32,18,31)(19,30,29,34)(21,35,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364083);