Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,15,30,13)(7,16,20,33,21,32,19,12)(8,22,25,28,26,29,24,9)(17,35,31,34,23,27,36,18);(1,2)(3,4)(5,7)(6,12)(8,23)(9,27)(10,20)(11,21)(13,32)(14,16)(15,33)(17,26)(18,22)(19,30)(24,31)(25,36)(28,35)(29,34);(2,4)(5,8,11,26)(6,9,15,28)(7,17)(10,24,30,25)(12,27)(13,22,14,29)(16,34)(18,32)(19,31)(20,36)(21,23)(33,35);(5,9,11,28)(6,8,15,26)(7,18,21,34)(10,29,30,22)(12,31,33,36)(13,25,14,24)(16,17,32,23)(19,35,20,27)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296229);