Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 | |
1944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,19)(7,20,25,53)(8,14,29,46)(9,18,11,16)(10,32,36,49)(13,38,42,24)(15,30,44,27)(17,31,51,35)(21,40,26,63)(22,58,61,47)(23,62,37,50)(28,64,59,48)(33,52,54,41)(34,45,65,68)(39,60,43,56)(55,67,57,66);(5,7,12,25)(6,13,19,42)(8,26,29,21)(9,23,11,37)(10,33,36,54)(14,43,46,39)(15,47,44,58)(16,41,18,52)(17,50,51,62)(20,49,53,32)(22,59,61,28)(24,31,38,35)(27,57,30,55)(34,60,65,56)(40,64,63,48)(45,66,68,67);(2,4)(5,8)(6,14)(7,21)(9,30)(10,28)(11,27)(12,29)(13,39)(15,16)(17,45)(18,44)(19,46)(20,40)(22,54)(23,57)(24,56)(25,26)(31,65)(32,64)(33,61)(34,35)(36,59)(37,55)(38,60)(41,58)(42,43)(47,52)(48,49)(50,67)(51,68)(53,63)(62,66);(5,8)(6,15)(7,22)(9,27)(10,34)(11,30)(12,29)(13,40)(14,16)(17,45)(18,46)(19,44)(20,39)(21,54)(23,60)(24,55)(25,61)(26,33)(28,35)(31,59)(32,48)(36,65)(37,56)(38,57)(41,66)(42,63)(43,53)(47,50)(49,64)(51,68)(52,67)(58,62);(5,9,12,11)(6,16,19,18)(7,23,25,37)(8,27,29,30)(10,31,36,35)(13,41,42,52)(14,44,46,15)(17,49,51,32)(20,50,53,62)(21,55,26,57)(22,60,61,56)(24,33,38,54)(28,34,59,65)(39,47,43,58)(40,66,63,67)(45,64,68,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^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,c^b=ci,d^b=df,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530841);