Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,22,24,30,26,29,25,27)(23,31,33,35,32,28,34,36);(2,4)(7,13)(10,17)(14,19)(18,20)(21,23)(22,28)(24,33)(25,34)(26,32)(27,35)(29,31)(30,36);(7,13)(10,17)(14,19)(18,20)(22,29)(23,32)(27,30)(33,34);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,30,29,27)(23,33,32,34)(28,36,31,35)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403644);