Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,18,11)(8,19)(9,22)(10,13)(12,21)(14,23)(16,29,30,33)(17,25,32,28)(20,34)(24,31,27,36)(26,35);(1,2)(3,4)(5,7,9,17,10,18,23,32)(6,11,22,28,13,15,14,25)(8,16,21,31,20,30,35,36)(12,24,34,29,26,27,19,33);(2,4)(5,8)(6,12,13,26)(7,16)(9,21)(10,20)(11,24,15,27)(14,19,22,34)(17,31)(18,30)(23,35)(25,33,28,29)(32,36);(8,20)(11,15)(12,26)(16,30)(19,34)(21,35)(25,28)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285507, #72285522, #72285631, … |
| maximal normal subgroup of | #72285507, #72285522, #72285631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357162);