Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,17,12,18,11,16)(7,19,22,13,23,39,36,40)(8,24,27,43,29,42,28,14)(9,30,33,46,35,45,34,15)(20,49,51,65,25,37,55,47)(21,52,54,48,31,38,63,61)(26,58,32,44,60,41,59,56)(50,67,53,68,57,66,62,64);(1,2)(3,4)(5,7,10,22,12,23,11,36)(6,13,16,19,18,40,17,39)(8,25,27,55,29,20,28,51)(9,31,33,63,35,21,34,54)(14,37,42,65,43,49,24,47)(15,38,45,48,46,52,30,61)(26,50,32,53,60,57,59,62)(41,64,44,66,58,68,56,67);(2,4)(5,8,12,29)(6,14,18,43)(7,20,23,25)(9,32,35,59)(10,28,11,27)(13,37,40,49)(15,44,46,56)(16,24,17,42)(19,47,39,65)(21,53,31,62)(22,55,36,51)(26,34,60,33)(30,58,45,41)(38,66,52,67)(48,64,61,68)(50,63,57,54);(5,9,11,34,12,35,10,33)(6,15,17,30,18,46,16,45)(7,21,36,63,23,31,22,54)(8,26,28,59,29,60,27,32)(13,38,39,61,40,52,19,48)(14,41,24,56,43,58,42,44)(20,50,55,62,25,57,51,53)(37,64,47,67,49,68,65,66)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383906);