Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3809 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)(7,11,16,25)(8,18,19,30)(9,14)(10,13)(12,20,27,22)(15,31)(17,26,21,29)(23,28)(24,34)(32,35)(33,36);(5,7)(6,11,13,25)(8,15)(9,21)(10,16)(12,24,27,36)(14,29,28,26)(17,23)(18,35,30,31)(19,32)(20,34)(22,33);(5,8)(6,12)(7,15)(9,22)(10,19)(11,24)(13,27)(14,30)(16,32)(17,34)(18,28)(20,23)(21,33)(25,36)(26,35)(29,31);(2,4)(7,16)(11,25)(15,32)(17,21)(24,36)(26,29)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287221, #72287295, #72287308, … |
| maximal normal subgroup of | #72287221, #72287295, #72287308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299885);