Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2722 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,29)(20,27)(21,26)(22,30)(23,32)(24,28)(25,31)(33,35)(34,36);(5,7,9,19,11,20,10,24)(6,12,15,17,16,28,14,27)(8,21,22,18,23,34,25,35)(13,29,31,26,32,33,30,36);(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,30)(15,31)(16,32)(17,33)(19,34)(20,35)(21,24)(27,29)(28,36);(5,9,11,10)(6,14,16,15)(7,19,20,24)(8,22,23,25)(12,27,28,17)(13,30,32,31)(18,34,35,21)(26,29,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313074, #72313078, #72313103, … |
| maximal normal subgroup of | #72313074, #72313078, #72313103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264980);