Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×C4⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1422 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)(5,6,10,16,11,17,22,15)(7,18,20,12,8,13,21,28)(9,24,25,14,26,30,36,31)(19,33,34,32,23,27,35,29);(2,4)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,34,26,23,36,35)(14,27,24,32,31,33,30,29);(2,4)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,26)(20,22)(24,27)(25,35)(30,33)(31,32)(34,36);(2,4)(5,9,10,25,11,26,22,36)(6,14,15,24,17,31,16,30)(7,19,20,34,8,23,21,35)(12,27,18,32,28,33,13,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268668);