Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1370 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)(7,16,20,30)(8,13,22,28)(9,11)(10,15)(12,17,26,19)(14,23)(18,33)(21,29,24,25)(27,32)(31,34)(35,36);(6,11)(7,17)(8,21)(9,23)(12,26)(13,28)(14,15)(18,34)(19,20)(22,24)(27,31)(33,36);(2,4)(5,7,9,17,10,20,23,19)(6,12,14,30,15,26,11,16)(8,18,24,32,22,34,21,35)(13,27,29,33,28,36,25,31);(2,4)(5,8,10,22)(6,13,15,28)(7,18,20,34)(9,24,23,21)(11,25,14,29)(12,27,26,36)(16,31,30,33)(17,32,19,35)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,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, 295585);