Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,14)(10,25)(11,15)(12,19)(17,32)(18,27)(20,31)(21,30)(22,28)(23,35)(24,29)(26,33)(34,36);(5,7,10,19,11,20,9,18)(6,12,14,16,15,27,25,31)(8,21,23,33,24,17,22,34)(13,26,28,30,29,36,35,32);(5,8)(6,13)(7,17,20,21)(9,22)(10,23)(11,24)(12,26,27,36)(14,28)(15,29)(16,30,31,32)(18,33,19,34)(25,35);(5,9,11,10)(6,14,15,25)(7,18,20,19)(8,22,24,23)(12,16,27,31)(13,28,29,35)(17,33,21,34)(26,30,36,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316334, #72316348, #72316355, … |
| maximal normal subgroup of | #72316334, #72316348, #72316355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265677);