Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33719 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,18,29)(8,20)(9,12)(10,24)(11,17,25,34)(13,22)(14,23)(16,30,19,35)(21,36)(26,33,27,32)(28,31);(5,7,8,17,10,19,22,33,9,18,21,34,23,16,31,32)(6,11,13,15,14,27,20,30,12,25,28,29,24,26,36,35);(1,2,3,4)(6,12)(11,25)(13,28)(14,24)(15,29)(20,36)(26,27)(30,35);(7,16,18,19)(8,21)(11,26,25,27)(13,28)(15,30,29,35)(17,33,34,32)(20,36)(22,31)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fi,d^2=g,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,g^b=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224815, #72224826, #72250863, … |
| maximal normal subgroup of | #72224815, #72224826, #72250863, … |
| maximal quotient of | #72224824, #72225276, #72225298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403322);