Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,27)(10,33)(11,14)(12,20)(13,25)(15,31)(16,30)(18,50)(19,49)(21,57)(23,60)(24,63)(26,46)(28,45)(29,44)(32,47)(34,43)(35,40)(36,62)(37,61)(38,54)(39,52)(41,51)(42,55)(48,53)(56,66)(58,59)(64,67)(65,68);(5,7,11,19)(6,12,14,38)(8,23,26,62)(9,28,32,61)(10,34,30,21)(13,39,40,50)(15,43,42,57)(16,45,33,37)(17,47,49,27)(18,51,52,24)(20,55,54,31)(22,59,46,48)(25,64,35,56)(29,58,65,53)(36,63,60,41)(44,67,68,66);(2,4)(5,6)(7,12)(8,13)(9,15)(10,16)(11,14)(17,20)(18,36)(19,38)(21,37)(22,25)(23,39)(24,41)(26,40)(27,31)(28,43)(29,44)(30,33)(32,42)(34,45)(35,46)(47,55)(48,56)(49,54)(50,62)(51,63)(52,60)(53,66)(57,61)(58,67)(59,64)(65,68);(5,8,11,26)(6,13,14,40)(7,18,19,52)(9,29,32,65)(10,35,30,25)(12,36,38,60)(15,44,42,68)(16,46,33,22)(17,48,49,59)(20,56,54,64)(21,58,34,53)(23,61,62,28)(24,55,51,31)(27,41,47,63)(37,67,45,66)(39,57,50,43);(6,14)(7,19)(13,40)(15,42)(16,33)(18,52)(20,54)(21,34)(22,46)(23,62)(27,47)(28,61)(41,63)(44,68)(53,58)(56,64)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530848);