Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| 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 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,14,28,42,9,15,7,30,5,12,27,41)(3,16,20,44,21,10,31,34,19,36,29,45,18,33,51,35)(4,11,23,38,22,37,52,61,25,39,24,55,26,40,53,62)(17,46,49,64,47,58,56,60,48,43,54,63,50,32,57,59);(1,3,7,20,8,21,27,31,9,19,6,29,5,18,28,51)(2,10,13,35,14,36,42,44,15,33,30,34,12,16,41,45)(4,17,24,49,22,47,53,56,25,48,23,54,26,50,52,57)(11,32,38,60,37,46,61,63,39,58,55,59,40,43,62,64);(1,4)(2,11)(3,17)(5,26)(6,23)(7,24)(8,22)(9,25)(10,32)(12,40)(13,38)(14,37)(15,39)(16,43)(18,50)(19,48)(20,49)(21,47)(27,53)(28,52)(29,54)(30,55)(31,56)(33,58)(34,59)(35,60)(36,46)(41,62)(42,61)(44,63)(45,64)(51,57)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,e^b=ei,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72319952, #72319981, #72319992, … |
| maximal normal subgroup of | #72319952, #72319981, #72319992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53597);