Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2228 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)(7,12,20,8)(9,14,24,31)(10,16)(11,15)(13,23,21,26)(17,18)(19,28)(22,33)(25,30,27,32)(29,36)(34,35);(1,2,3,4)(5,7,17,21,10,20,11,13)(6,12,18,26,16,8,15,23)(9,19,25,34,24,33,27,29)(14,28,30,35,31,22,32,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,12)(11,26)(14,29)(15,20)(16,21)(17,23)(19,30)(24,28)(25,36)(27,35)(31,34)(32,33);(5,9)(6,14)(7,19)(8,22)(10,24)(11,27)(12,28)(13,29)(15,32)(16,31)(17,25)(18,30)(20,33)(21,34)(23,36)(26,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273821);