Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 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)(5,6,11,16,12,18,10,17)(7,19,22,34,23,33,21,13)(8,24,26,36,27,32,25,14)(9,28,20,31,30,35,29,15);(2,4)(5,7)(6,13,18,34)(8,9)(10,21)(11,22)(12,23)(14,35,36,28)(15,32,31,24)(16,19,17,33)(20,26)(25,29)(27,30);(2,4)(5,8)(6,14)(7,20)(9,21)(10,25)(11,26)(12,27)(13,31)(15,34)(16,24)(17,32)(18,36)(19,35)(22,30)(23,29)(28,33);(2,4)(5,9,10,29,12,30,11,20)(6,15,16,28,18,31,17,35)(7,8,21,25,23,27,22,26)(13,32,19,14,34,24,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279973);