Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,19)(8,23)(9,15)(10,16)(11,17)(12,18)(13,22)(14,27)(20,35)(21,30)(24,34)(25,29)(26,32)(28,33)(31,36);(5,7,11,22,12,9,10,21)(6,13,16,19,18,30,17,15)(8,24,27,25,28,36,26,20)(14,31,23,29,32,34,33,35);(2,4)(5,8,11,27,12,28,10,26)(6,14,16,23,18,32,17,33)(7,20,22,24,9,25,21,36)(13,29,19,34,30,35,15,31);(5,9)(6,15)(7,12)(8,25)(10,22)(11,21)(13,16)(14,31)(17,30)(18,19)(20,28)(23,29)(24,26)(27,36)(32,34)(33,35);(5,10,12,11)(6,16,18,17)(7,21,9,22)(8,26,28,27)(13,19,30,15)(14,23,32,33)(20,36,25,24)(29,34,35,31)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346215);