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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1287 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)(11,19)(12,28)(13,23)(14,21)(15,36)(16,20)(17,26)(18,32)(22,37)(24,49)(25,39)(27,56)(29,48)(30,60)(31,59)(33,58)(34,46)(35,41)(38,66)(40,50)(42,53)(43,64)(44,63)(45,61)(47,54)(51,71)(52,57)(55,65)(62,69)(67,72)(68,70);(1,3)(2,5)(4,8)(6,7)(9,11,14,26)(10,17,21,19)(12,24,31,52)(13,34,16,25)(15,27,35,55)(18,38,44,42)(20,46,23,39)(22,40,48,47)(28,57,59,49)(29,50,37,54)(30,62,33,51)(32,53,63,66)(36,65,41,56)(43,68,45,67)(58,69,60,71)(61,70,64,72);(9,12,29,61,16,33,15,32,14,31,37,64,13,30,35,63)(10,18,41,58,23,45,22,28,21,44,36,60,20,43,48,59)(11,24,50,70,25,51,27,53,26,52,54,72,34,62,55,66)(17,38,56,69,39,67,40,57,19,42,65,71,46,68,47,49);(3,8)(5,7)(9,11)(10,19)(12,24)(13,34)(14,26)(15,27)(16,25)(17,21)(18,42)(20,39)(22,47)(23,46)(28,49)(29,50)(30,62)(31,52)(32,53)(33,51)(35,55)(36,56)(37,54)(38,44)(40,48)(41,65)(43,67)(45,68)(57,59)(58,71)(60,69)(61,70)(63,66)(64,72);(2,6)(5,7)(9,13,14,16)(10,20,21,23)(11,25,26,34)(12,30,31,33)(15,29,35,37)(17,39,19,46)(18,43,44,45)(22,41,48,36)(24,51,52,62)(27,54,55,50)(28,58,59,60)(32,61,63,64)(38,67,42,68)(40,65,47,56)(49,69,57,71)(53,72,66,70)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,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^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532057);