Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #386 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10714 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,14,29,42,9,15,7,31,5,12,28,41)(3,16,20,44,22,33,51,60,18,10,21,34,19,35,50,61)(4,23,25,54,24,39,56,38,26,40,30,55,27,11,36,37)(17,45,48,63,47,32,57,58,46,53,52,64,49,43,62,59);(1,3,7,21,8,22,28,50,9,18,6,20,5,19,29,51)(2,10,13,34,14,35,42,61,15,16,31,44,12,33,41,60)(4,17,25,48,24,47,56,57,26,46,30,52,27,49,36,62)(11,32,38,59,23,53,55,63,39,43,37,58,40,45,54,64);(1,4)(2,11)(3,17)(5,27)(6,30)(7,25)(8,24)(9,26)(10,32)(12,40)(13,38)(14,23)(15,39)(16,43)(18,46)(19,49)(20,52)(21,48)(22,47)(28,56)(29,36)(31,37)(33,45)(34,59)(35,53)(41,54)(42,55)(44,58)(50,57)(51,62)(60,64)(61,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,d^b=di,e^c=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72341758 |
| maximal normal subgroup of | #72341758 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53612);