Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10,14,20)(11,23,26,17)(12,29,31,63)(13,22,16,19)(15,35,34,46)(18,43,45,32)(21,33,47,37)(24,52,54,62)(25,40,28,42)(27,58,57,66)(30,64,38,48)(36,49,65,44)(39,55,50,70)(41,60,51,56)(53,68,61,71)(59,67,72,69);(1,2)(3,6)(4,5)(7,8)(9,11,14,26)(10,17,20,23)(12,24,31,54)(13,25,16,28)(15,27,34,57)(18,39,45,50)(19,40,22,42)(21,41,47,51)(29,62,63,52)(30,53,38,61)(32,55,43,70)(33,56,37,60)(35,66,46,58)(36,59,65,72)(44,67,49,69)(48,68,64,71);(9,12)(10,18)(11,24)(13,30)(14,31)(15,36)(16,38)(17,39)(19,44)(20,45)(21,48)(22,49)(23,50)(25,53)(26,54)(27,59)(28,61)(29,35)(32,37)(33,43)(34,65)(40,67)(41,68)(42,69)(46,63)(47,64)(51,71)(52,58)(55,60)(56,70)(57,72)(62,66);(2,5)(6,8)(9,13,14,16)(10,19,20,22)(11,25,26,28)(12,30,31,38)(15,33,34,37)(17,40,23,42)(18,44,45,49)(21,46,47,35)(24,53,54,61)(27,56,57,60)(29,48,63,64)(32,36,43,65)(39,67,50,69)(41,58,51,66)(52,71,62,68)(55,59,70,72);(1,3,4,7)(2,6,5,8)(10,20)(11,26)(18,45)(19,22)(21,47)(24,54)(25,28)(27,57)(29,63)(35,46)(44,49)(48,64)(53,61)(55,70)(56,60)(59,72)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531845);