Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25,27,36)(16,30,34,40)(17,23)(18,24)(19,22)(20,28,37,29)(21,33,39,32)(26,42)(31,41)(35,43)(38,44);(2,4)(6,9)(10,12)(13,15)(14,20)(16,31)(17,29)(18,28)(19,27)(21,38)(22,37)(23,36)(24,25)(26,32)(30,42)(33,43)(34,44)(35,40)(39,41);(2,4)(13,16)(14,21)(15,26)(17,33)(18,32)(19,34)(20,35)(22,39)(23,30)(24,40)(25,41)(27,43)(28,44)(29,31)(36,38)(37,42);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(21,39)(23,24)(26,43)(28,29)(30,40)(31,44)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265892);