Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 | |
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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,35)(15,37)(16,21)(18,31)(20,29)(22,47)(23,39)(24,40)(25,53)(26,42)(28,55)(30,56)(32,63)(33,58)(34,46)(36,64)(38,51)(41,48)(43,59)(44,61)(45,62)(49,69)(50,70)(52,68)(54,67)(57,60)(65,72)(66,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,28)(13,23)(14,25)(15,24)(16,26)(18,43)(19,39)(20,41)(21,42)(22,33)(27,55)(29,48)(30,50)(31,59)(32,49)(34,52)(35,53)(36,54)(37,40)(38,45)(44,66)(46,68)(47,58)(51,62)(56,70)(57,65)(60,72)(61,71)(63,69)(64,67);(9,12,29,60,15,32,36,62,16,33,34,61,13,30,14,31)(10,18,35,56,19,44,46,58,21,45,64,63,37,57,20,27)(11,22,48,71,24,50,54,59,26,28,52,72,23,49,25,51)(17,38,53,69,39,65,68,55,42,43,67,70,40,66,41,47);(3,8)(5,7)(9,13,16,15)(10,19,21,37)(11,23,26,24)(12,30,33,32)(14,34,36,29)(17,39,42,40)(18,44,45,57)(20,35,46,64)(22,49,28,50)(25,52,54,48)(27,56,58,63)(31,61,62,60)(38,65,43,66)(41,53,68,67)(47,69,55,70)(51,72,59,71);(2,6)(5,7)(9,13,16,15)(10,19,21,37)(11,24,26,23)(12,30,33,32)(14,34,36,29)(17,40,42,39)(18,44,45,57)(20,35,46,64)(22,50,28,49)(25,48,54,52)(27,56,58,63)(31,61,62,60)(38,66,43,65)(41,67,68,53)(47,70,55,69)(51,71,59,72)Polycyclic presentation:
2^9|c^2=gh,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,c^b=ci,d^b=df,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532037);