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 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17,12,16,10,18)(7,19,22,30,9,13,21,15)(8,23,28,32,26,33,27,14)(20,35,24,29,25,31,36,34);(1,2)(3,4)(5,7,12,9)(6,13,16,19)(8,24,26,36)(10,22,11,21)(14,31,32,35)(15,17,30,18)(20,27,25,28)(23,29,33,34);(2,4)(5,8)(6,14,16,32)(7,20)(9,25)(10,27)(11,28)(12,26)(13,29,19,34)(15,31,30,35)(17,23,18,33)(21,36)(22,24);(5,9,12,7)(6,15,16,30)(8,25,26,20)(10,21,11,22)(13,18,19,17)(14,31,32,35)(23,29,33,34)(24,27,36,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296193);