Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629034 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2016 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)(3,6)(4,5)(7,8)(9,10,16,22)(11,17,25,39)(12,28,32,43)(13,33,35,46)(14,21,15,20)(18,31,44,30)(19,36,45,34)(23,47,49,63)(24,52,53,65)(26,41,27,40)(29,57,59,42)(37,51,62,50)(38,55,64,54)(48,68,69,61)(56,60,66,58)(67,71,72,70);(2,5)(6,8)(9,11,16,25)(10,17,22,39)(12,23,32,49)(13,24,35,53)(14,26,15,27)(18,37,44,62)(19,38,45,64)(20,40,21,41)(28,47,43,63)(29,48,59,69)(30,50,31,51)(33,52,46,65)(34,54,36,55)(42,61,57,68)(56,67,66,72)(58,70,60,71);(9,12,15,31,16,32,14,30)(10,18,21,43,22,44,20,28)(11,23,27,51,25,49,26,50)(13,29,36,60,35,59,34,58)(17,37,41,63,39,62,40,47)(19,42,46,66,45,57,33,56)(24,48,55,71,53,69,54,70)(38,61,65,72,64,68,52,67);(9,13)(10,19)(11,24)(12,29)(14,34)(15,36)(16,35)(17,38)(18,42)(20,33)(21,46)(22,45)(23,48)(25,53)(26,54)(27,55)(28,56)(30,58)(31,60)(32,59)(37,61)(39,64)(40,52)(41,65)(43,66)(44,57)(47,67)(49,69)(50,70)(51,71)(62,68)(63,72);(1,3,4,7)(2,6,5,8)(11,25)(17,39)(23,49)(24,53)(26,27)(37,62)(38,64)(40,41)(47,63)(48,69)(50,51)(52,65)(54,55)(61,68)(67,72)(70,71)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,e^b=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537558);