Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(8,21)(9,25)(10,24)(13,31)(14,15)(16,26)(19,23)(20,36)(22,28)(27,29)(30,32)(33,35);(1,2)(3,4)(5,6)(7,18)(8,22)(9,26)(10,28)(11,14)(12,21)(13,19)(15,25)(16,24)(17,35)(20,34)(23,30)(27,33)(29,31)(32,36);(2,4)(5,7)(6,13)(8,23)(9,27)(10,29)(11,20)(12,30)(14,31)(15,33)(16,34)(17,25)(18,28)(19,24)(21,36)(22,35)(26,32);(6,14)(12,15)(13,31)(16,28)(18,34)(22,26)(30,33)(32,35)Polycyclic presentation:
2^9|e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250029, #72250041, #72250044, … |
| maximal normal subgroup of | #72250029, #72250041, #72250044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406074);