Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219498 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1296 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)(7,18)(8,21,26,35)(9,20)(10,28)(11,16)(12,17)(13,15)(14,25,32,24)(19,30,33,34)(22,23,36,31)(27,29);(1,2)(3,4)(5,7,10,15,12,20,11,29)(6,13,16,18,17,27,28,9)(8,22,25,19,26,36,24,33)(14,31,35,30,32,23,21,34);(2,4)(5,8)(6,14,17,32)(7,19,20,33)(9,23)(10,24)(11,25)(12,26)(13,30)(15,22,29,36)(16,21,28,35)(18,31)(27,34);(5,9,10,13,12,18,11,27)(6,15,16,20,17,29,28,7)(8,23,24,30,26,31,25,34)(14,22,21,33,32,36,35,19)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272967);