Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,28)(23,26,31,34)(24,27,30,33)(29,32,35,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(23,29)(24,30)(26,32)(27,33)(31,35)(34,36);(2,4)(21,23,24,29,25,31,30,35)(22,26,27,32,28,34,33,36);(6,11)(10,13)(12,19)(18,20);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,23,30,35,25,31,24,29)(22,26,33,36,28,34,27,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716729);