Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2610 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)(9,10,14,20,16,22,15,21)(11,17,26,41,25,40,37,48)(12,27,30,45,32,44,31,18)(13,33,34,46,35,47,36,19)(23,42,28,56,50,65,55,38)(24,51,53,63,52,64,54,39)(29,43,59,67,60,68,61,57)(49,62,70,72,69,71,58,66);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,28)(13,24)(14,26)(15,37)(16,25)(18,42)(19,39)(20,41)(21,48)(22,40)(23,31)(27,56)(29,58)(30,50)(32,55)(33,51)(34,53)(35,52)(36,54)(38,44)(43,66)(45,65)(46,63)(47,64)(49,59)(57,71)(60,70)(61,69)(62,67)(68,72);(9,12,16,32)(10,18,22,45)(11,23,25,50)(13,29,35,60)(14,31,15,30)(17,38,40,56)(19,43,46,68)(20,44,21,27)(24,49,52,69)(26,55,37,28)(33,57,47,67)(34,61,36,59)(39,62,63,71)(41,65,48,42)(51,66,64,72)(53,58,54,70);(2,6)(3,8)(9,13)(10,19)(11,24)(12,29)(14,36)(15,34)(16,35)(17,39)(18,43)(20,47)(21,33)(22,46)(23,49)(25,52)(26,54)(27,57)(28,58)(30,61)(31,59)(32,60)(37,53)(38,62)(40,63)(41,64)(42,66)(44,67)(45,68)(48,51)(50,69)(55,70)(56,71)(65,72);(11,25)(17,40)(23,50)(24,52)(26,37)(28,55)(38,56)(39,63)(41,48)(42,65)(49,69)(51,64)(53,54)(58,70)(62,71)(66,72)Polycyclic presentation:
2^9|a^2=g,c^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410510);