Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,26)(24,28)(25,27)(29,33)(30,34)(31,35)(32,36);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23,25,31)(22,26,27,35)(24,29,32,30)(28,33,36,34);(2,4)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19);(7,15)(10,13)(14,20)(18,19)(22,27)(23,29)(24,32)(26,34)(30,31)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579220);