Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623311 |
| 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 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1599 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)(9,10,16,21)(11,17,24,40)(12,27,30,60)(13,19,15,36)(14,32,35,61)(18,42,45,29)(20,31,47,34)(22,48,50,70)(23,39,26,57)(25,54,53,71)(28,59,37,46)(33,43,62,44)(38,63,65,51)(41,52,67,56)(49,69,58,68)(55,64,72,66);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,38)(19,39)(20,41)(21,40)(27,48)(28,49)(29,51)(30,50)(31,52)(32,54)(33,55)(34,56)(35,53)(36,57)(37,58)(42,63)(43,64)(44,66)(45,65)(46,68)(47,67)(59,69)(60,70)(61,71)(62,72);(3,8)(5,7)(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,43)(20,46)(21,45)(23,49)(24,50)(25,55)(26,58)(27,32)(29,34)(31,42)(35,62)(36,44)(39,64)(40,65)(41,68)(47,59)(48,54)(51,56)(52,63)(53,72)(57,66)(60,61)(67,69)(70,71);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,28,30,37)(14,31,35,34)(17,39,40,57)(18,43,45,44)(20,32,47,61)(22,49,50,58)(25,52,53,56)(27,59,60,46)(29,33,42,62)(38,64,65,66)(41,54,67,71)(48,69,70,68)(51,55,63,72);(11,24)(17,40)(22,50)(23,26)(25,53)(38,65)(39,57)(41,67)(48,70)(49,58)(51,63)(52,56)(54,71)(55,72)(64,66)(68,69)Polycyclic presentation:
2^9|a^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530512);