Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,16,9,13)(7,17,20,28,21,27,19,12)(8,22,25,30,23,29,24,14)(18,32,35,36,33,26,34,31);(1,3,4,2)(5,7,10,20,11,21,9,19)(6,12,13,27,16,28,15,17)(8,18,25,35,23,33,24,34)(14,26,29,36,30,32,22,31);(6,13,16,15)(8,23)(12,17,28,27)(14,22,30,29)(18,33)(24,25)(26,36,32,31)(34,35);(5,8,9,24,11,23,10,25)(6,14,15,22,16,30,13,29)(7,18,19,34,21,33,20,35)(12,26,17,31,28,32,27,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=e,e^2=i,f^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,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250366, #72250373, #72250386, … |
| maximal normal subgroup of | #72250366, #72250373, #72250386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277228);