Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2087 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,11,19,13,18,12,36)(7,20,24,46,26,50,25,38)(8,27,30,59,32,58,31,42)(9,17,33,45,34,47,10,16)(14,23,40,56,39,35,49,22)(15,28,44,60,43,61,63,29)(21,41,53,67,55,37,54,64)(48,68,66,62,65,51,57,52);(1,2)(3,4)(5,7,10,35,12,25,34,56,13,26,33,23,11,24,9,22)(6,14,17,46,19,40,45,50,18,39,47,38,36,49,16,20)(8,21,29,62,31,54,61,68,32,55,60,52,30,53,28,51)(15,41,27,57,44,67,59,48,43,37,58,66,63,64,42,65);(2,4)(5,8,9,28,11,30,33,60,13,32,34,61,12,31,10,29)(6,15,16,42,36,63,47,58,18,43,45,59,19,44,17,27)(7,21,22,51,24,53,23,52,26,55,56,68,25,54,35,62)(14,37,20,48,49,67,38,57,39,41,50,65,40,64,46,66)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56094);