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 | C2×#3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,14)(9,26)(10,28)(11,16)(12,20)(13,22)(15,24)(18,32)(19,35)(21,36)(23,34)(25,33)(27,31)(29,30);(1,2)(3,4)(7,18)(8,22)(9,25)(10,24)(12,30)(13,14)(15,28)(17,32)(19,23)(20,29)(21,27)(26,33)(31,36)(34,35);(1,3,4,2)(5,7)(6,12)(8,23)(9,27)(10,29)(11,19)(13,30)(14,17)(15,34)(16,31)(18,25)(20,22)(21,24)(26,36)(28,32)(33,35);(7,19)(12,31)(17,35)(18,23)(20,27)(21,29)(30,36)(32,34)Polycyclic presentation:
2^9|c^2=e,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 | #72250359, #72250370, #72250390, … |
| maximal normal subgroup of | #72250359, #72250370, #72250390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406282);