Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473174 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1870 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)(3,8)(4,6)(5,7)(9,10,16,22)(11,23,27,48)(12,28,30,43)(13,19,34,45)(14,21,15,20)(17,37,40,26)(18,31,42,32)(24,51,53,38)(25,50,56,68)(29,58,33,62)(35,47,36,46)(39,63,66,57)(41,59,44,60)(49,61,67,54)(52,65,55,64)(69,72,70,71);(3,4)(6,8)(9,11)(10,17)(12,24)(13,25)(14,37)(15,26)(16,27)(18,38)(19,39)(20,48)(21,23)(22,40)(28,49)(29,52)(30,53)(31,61)(32,54)(33,55)(34,56)(35,63)(36,57)(41,64)(42,51)(43,67)(44,65)(45,66)(46,68)(47,50)(58,69)(59,72)(60,71)(62,70);(1,3,5,4)(2,6,7,8)(9,12,14,31,16,30,15,32)(10,18,20,43,22,42,21,28)(11,24,37,61,27,53,26,54)(13,33,35,60,34,29,36,59)(17,38,48,67,40,51,23,49)(19,44,46,58,45,41,47,62)(25,55,63,71,56,52,57,72)(39,65,68,69,66,64,50,70);(1,4,5,3)(2,6,7,8)(9,13,16,34)(10,19,22,45)(11,25,27,56)(12,29,30,33)(14,35,15,36)(17,39,40,66)(18,41,42,44)(20,46,21,47)(23,50,48,68)(24,52,53,55)(26,57,37,63)(28,58,43,62)(31,59,32,60)(38,64,51,65)(49,69,67,70)(54,71,61,72);(2,7)(6,8)(12,30)(13,34)(18,42)(19,45)(24,53)(25,56)(28,43)(31,32)(35,36)(38,51)(39,66)(46,47)(49,67)(50,68)(54,61)(57,63)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390501);