Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
4012 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)(7,16)(8,14)(9,21)(10,25)(11,13)(12,18)(15,23)(17,33)(19,32)(20,30)(22,28)(24,35)(26,31)(27,34)(29,36);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,28)(15,31)(16,21)(18,23)(20,34)(24,36)(25,29)(30,33)(32,35);(2,4);(6,13)(12,27)(14,25)(15,30)(16,32)(21,35)(28,29)(31,33);(5,8,11,10)(6,14,13,25)(7,17,19,26)(9,20,24,23)(12,28,27,29)(15,21,30,35)(16,33,32,31)(18,22,34,36)Polycyclic presentation:
2^9|e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,h^a=hi,f^b=fh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249895, #72249903, #72249915, … |
| maximal normal subgroup of | #72249895, #72249903, #72249915, … |
| maximal quotient of | #72265737, #72285609, C4×#33517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530008);