Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14572 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,15,18,8,19,25,17,13)(5,14,23,20,22,9,21,26)(16,28,29,24,31,32,30,27);(1,3)(2,7)(4,8)(5,20)(6,10)(9,23)(11,12)(13,19)(14,21)(15,17)(16,28)(18,25)(22,26)(24,29)(27,30)(31,32);(1,4,7,19)(2,8)(3,13)(5,16,22,31)(6,18,12,17)(9,24)(10,15)(11,25)(14,27)(20,28)(21,30,23,29)(26,32);(1,5)(2,9)(3,14)(4,16)(6,23)(7,22)(8,24)(10,20)(11,26)(12,21)(13,27)(15,28)(17,30)(18,29)(19,31)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=egh,e^2=i,f^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,e^b=ei,f^b=fi,e^c=ei,g^c=gi,e^d=ei,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, 297648);