Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,14)(9,13)(10,28)(11,17)(15,25)(16,24)(18,35)(19,30)(20,32)(21,36)(22,29)(23,33)(26,31)(27,34);(1,2)(3,4)(5,7)(6,12)(8,19)(9,26)(10,18)(11,23)(13,31)(14,30)(15,32)(16,29)(17,33)(20,25)(21,27)(22,24)(28,35)(34,36);(1,3,4,2)(6,13)(7,18)(8,24)(9,25)(10,27)(12,29)(14,28)(15,16)(17,34)(19,21)(20,23)(22,26)(30,33)(32,35);(5,8,11,24)(6,14,17,16)(7,19,23,22)(9,25,27,10)(12,30,33,29)(13,15,34,28)(18,26,20,21)(31,32,36,35)Polycyclic presentation:
2^9|c^2=e,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=di,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321674, #72321689, #72321715, … |
| maximal normal subgroup of | #72321674, #72321689, #72321715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406284);