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×#3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1146 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)(7,16)(8,21)(9,10)(12,27)(13,29)(14,26)(17,32)(18,33)(19,23)(20,34)(22,35)(24,25)(28,36)(30,31);(1,2)(3,4)(5,6)(7,17)(8,22)(9,26)(10,14)(11,15)(12,19)(13,25)(16,32)(18,31)(20,28)(21,35)(23,27)(24,29)(30,33)(34,36);(1,3,4,2)(5,7)(6,12)(8,23)(9,20)(10,19)(11,18)(13,30)(14,31)(15,28)(16,21)(17,26)(22,32)(24,33)(25,34)(27,29)(35,36);(7,18)(12,28)(16,33)(17,31)(19,20)(23,34)(27,36)(30,32)Polycyclic presentation:
2^9|c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250274, #72250303, #72250324, … |
| maximal normal subgroup of | #72250274, #72250303, #72250324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406216);