Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,33)(15,36)(16,21)(18,31)(20,34)(22,47)(23,38)(24,39)(25,54)(26,41)(28,46)(29,56)(30,57)(32,58)(35,42)(37,50)(40,53)(43,59)(44,60)(45,61)(48,66)(49,67)(51,72)(52,68)(55,62)(63,69)(64,71)(65,70);(1,3)(2,5)(4,8)(6,7)(9,11,16,26)(10,17,21,41)(12,22,32,52)(13,23,15,24)(14,25,35,55)(18,37,45,64)(19,38,36,39)(20,40,46,66)(27,47,58,68)(28,48,34,53)(29,49,30,51)(31,50,61,71)(33,54,42,62)(43,63,44,65)(56,67,57,72)(59,69,60,70);(9,12,28,59,15,30,14,31,16,32,34,60,13,29,35,61)(10,18,42,57,19,43,20,27,21,45,33,56,36,44,46,58)(11,22,48,69,24,51,25,50,26,52,53,70,23,49,55,71)(17,37,62,72,38,63,40,47,41,64,54,67,39,65,66,68);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,26,24)(12,29,32,30)(14,28,35,34)(17,38,41,39)(18,43,45,44)(20,33,46,42)(22,49,52,51)(25,48,55,53)(27,56,58,57)(31,59,61,60)(37,63,64,65)(40,54,66,62)(47,67,68,72)(50,69,71,70);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,24,26,23)(12,30,32,29)(14,28,35,34)(17,39,41,38)(18,44,45,43)(20,33,46,42)(22,49,52,51)(25,53,55,48)(27,57,58,56)(31,60,61,59)(37,63,64,65)(40,62,66,54)(47,67,68,72)(50,69,71,70)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532111);