Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #38950 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,30)(23,29,33,27)(24,28,26,31)(32,36,34,35);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(23,32)(24,26)(27,35)(28,31)(29,36)(33,34);(1,2,3,4)(21,23,25,33)(22,27,30,29)(24,32,26,34)(28,35,31,36);(6,11)(10,13)(12,19)(18,20)(21,24,25,26)(22,28,30,31)(23,32,33,34)(27,35,29,36);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,23,25,33)(22,29,30,27)(24,32,26,34)(28,36,31,35)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6593379);