Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2412 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)(7,12,19,32)(8,21)(9,14,26,33)(10,16)(11,29)(13,24)(15,28)(17,30,22,27)(18,31)(20,36,34,23)(25,35);(2,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,26)(14,31)(15,23)(16,32)(17,25)(21,30)(24,33)(27,35)(28,34)(29,36);(5,8,11,24,10,25,28,31)(6,13,15,21,16,18,29,35)(7,17,34,33,19,22,20,14)(9,23,30,12,26,36,27,32);(2,4)(5,9,10,26)(6,14,16,33)(7,18,19,13)(8,23,25,36)(11,30,28,27)(12,31,32,24)(15,17,29,22)(20,21,34,35)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281229);