Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 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)(5,6)(7,18)(8,14,24,33)(9,13)(10,27)(11,17)(12,16)(15,20)(19,31,35,30)(21,28,22,36)(23,34,25,32)(26,29);(1,2,4,3)(5,7,12,9,11,20,10,26)(6,13,16,15,17,29,27,18)(8,21,25,35,24,22,23,19)(14,30,32,36,33,31,34,28);(5,8)(6,14)(7,19)(9,22)(10,25)(11,24)(12,23)(13,28)(15,31)(16,34)(17,33)(18,30)(20,35)(21,26)(27,32)(29,36);(1,3,4,2)(5,9,12,20,11,26,10,7)(6,15,16,29,17,18,27,13)(8,22,23,35,24,21,25,19)(14,31,34,36,33,30,32,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280175, #72280601, #72280644, … |
| maximal normal subgroup of | #72280175, #72280601, #72280644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285340);