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 | |
2520 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)(7,13)(8,21)(9,14)(10,15)(11,16)(12,17)(18,27)(19,32)(20,28)(22,35)(23,29)(24,30)(25,33)(26,31)(34,36);(2,4)(5,7)(6,13)(8,10)(9,23)(11,19)(12,20)(14,29)(15,21)(16,32)(17,28)(18,24)(22,26)(25,34)(27,30)(31,35)(33,36);(5,8,11,7,12,19,10,20)(6,13,15,21,17,28,16,32)(9,22,25,18,26,23,24,34)(14,27,30,35,31,36,33,29);(5,9)(6,14)(7,18)(8,22)(10,24)(11,25)(12,26)(13,27)(15,30)(16,33)(17,31)(19,23)(20,34)(21,35)(28,36)(29,32);(5,10,12,11)(6,15,17,16)(7,8,20,19)(9,24,26,25)(13,21,28,32)(14,30,31,33)(18,22,34,23)(27,35,36,29)Polycyclic presentation:
2^9|a^2=f,c^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,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, 6350143);