Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 | |
4886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,19)(8,25)(9,15)(10,16)(11,17)(12,18)(13,22)(14,21)(20,30)(23,32)(24,31)(26,36)(27,35)(28,34)(29,33);(1,2,3,4)(5,7,10,22,12,24,11,23)(6,13,17,19,18,32,16,31)(8,26,21,20,28,29,27,9)(14,33,25,30,35,36,34,15);(5,8,11,21,12,28,10,27)(6,14,16,25,18,35,17,34)(7,20,23,29,24,9,22,26)(13,30,31,36,32,15,19,33);(5,9)(6,15)(7,21)(8,22)(10,29)(11,26)(12,20)(13,25)(14,19)(16,33)(17,36)(18,30)(23,28)(24,27)(31,35)(32,34);(5,10,12,11)(6,16,18,17)(7,22,24,23)(8,27,28,21)(9,29,20,26)(13,31,32,19)(14,25,35,34)(15,33,30,36)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350476);