Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3062 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,6)(4,7)(9,10,16,20,15,22,14,21)(11,23,27,17,26,50,28,42)(12,29,32,18,34,52,33,45)(13,19,37,51,39,49,38,48)(24,40,60,53,30,43,56,69)(25,54,59,41,58,46,35,65)(31,61,64,44,36,66,63,47)(55,68,67,71,57,70,62,72);(1,2,5,6)(3,4,8,7)(9,11)(10,17)(12,30)(13,35)(14,28)(15,26)(16,27)(18,43)(19,46)(20,50)(21,23)(22,42)(24,34)(25,37)(29,53)(31,62)(32,56)(33,60)(36,67)(38,58)(39,59)(40,45)(41,48)(44,71)(47,72)(49,54)(51,65)(52,69)(55,64)(57,63)(61,68)(66,70);(1,3)(2,7)(4,6)(5,8)(9,12)(10,18)(11,24)(13,36)(14,32)(15,34)(16,33)(17,40)(19,47)(20,29)(21,52)(22,45)(23,53)(25,57)(26,30)(27,56)(28,60)(31,39)(35,62)(37,64)(38,63)(41,70)(42,43)(44,49)(46,71)(48,61)(50,69)(51,66)(54,72)(55,58)(59,67)(65,68);(2,6)(3,8)(9,13)(10,19)(11,25)(12,31)(14,38)(15,39)(16,37)(17,41)(18,44)(20,51)(21,48)(22,49)(23,54)(24,55)(26,58)(27,59)(28,35)(29,61)(30,57)(32,64)(33,63)(34,36)(40,68)(42,65)(43,70)(45,47)(46,50)(52,66)(53,71)(56,62)(60,67)(69,72);(1,4,5,7)(2,3,6,8)(11,26)(17,42)(23,50)(24,30)(25,58)(27,28)(35,59)(40,43)(41,65)(46,54)(53,69)(55,57)(56,60)(62,67)(68,70)(71,72)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089212);