Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219508 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1403 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)(7,15)(8,19,23,12)(9,24)(10,14)(11,17)(13,27)(16,33,20,32)(18,31)(21,22,30,26)(25,29)(28,35,36,34);(5,7,10,18)(6,11,14,29)(8,20,23,16)(9,25,27,17)(12,28,19,36)(13,31,24,15)(21,33,30,32)(22,34,26,35);(2,4)(5,8)(6,12)(7,16)(9,26)(10,23)(11,28)(13,21)(14,19)(15,32)(17,35)(18,20)(22,27)(24,30)(25,34)(29,36)(31,33);(5,6)(7,11)(8,21,23,30)(9,13)(10,14)(12,26,19,22)(15,17)(16,32,20,33)(18,29)(24,27)(25,31)(28,35,36,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288672);