Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 | |
1270 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)(6,8)(9,10,16,20)(11,17,24,41)(12,27,30,60)(13,19,15,36)(14,32,35,61)(18,42,44,29)(21,31,46,34)(22,48,50,70)(23,39,26,57)(25,54,53,71)(28,59,37,47)(33,43,62,45)(38,63,66,51)(40,52,68,56)(49,69,58,67)(55,64,72,65);(1,2)(3,6)(4,5)(7,8)(9,11,16,24)(10,17,20,41)(12,22,30,50)(13,23,15,26)(14,25,35,53)(18,38,44,66)(19,39,36,57)(21,40,46,68)(27,48,60,70)(28,49,37,58)(29,51,42,63)(31,52,34,56)(32,54,61,71)(33,55,62,72)(43,64,45,65)(47,67,59,69);(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,43)(20,44)(21,47)(23,49)(24,50)(25,55)(26,58)(27,32)(29,34)(31,42)(35,62)(36,45)(39,64)(40,67)(41,66)(46,59)(48,54)(51,56)(52,63)(53,72)(57,65)(60,61)(68,69)(70,71);(2,5)(6,8)(9,13,16,15)(10,19,20,36)(11,23,24,26)(12,28,30,37)(14,31,35,34)(17,39,41,57)(18,43,44,45)(21,32,46,61)(22,49,50,58)(25,52,53,56)(27,59,60,47)(29,33,42,62)(38,64,66,65)(40,54,68,71)(48,69,70,67)(51,55,63,72);(1,3,4,7)(2,6,5,8)(10,20)(11,24)(18,44)(19,36)(21,46)(22,50)(23,26)(25,53)(27,60)(32,61)(43,45)(47,59)(49,58)(51,63)(52,56)(55,72)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531311);