Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6502937 |
| 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 | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(9,10,15,20,16,21,14,35)(11,22,26,39,27,47,25,17)(12,28,30,42,31,41,36,18)(13,19,34,44,32,43,33,59)(23,37,49,55,50,62,54,45)(24,46,53,63,51,67,52,38)(29,56,58,64,57,65,60,40)(48,61,69,71,68,72,70,66);(1,2)(3,7)(4,6)(5,8)(9,11,15,26,16,27,14,25)(10,17,35,47,21,39,20,22)(12,23,36,54,31,50,30,49)(13,24,34,53,32,51,33,52)(18,37,28,55,42,62,41,45)(19,38,59,67,43,63,44,46)(29,48,60,70,57,68,58,69)(40,61,56,71,64,72,65,66);(1,3)(2,5)(4,8)(6,7)(9,12,16,31)(10,18,21,42)(11,23,27,50)(13,29,32,57)(14,30,15,36)(17,37,39,62)(19,40,43,64)(20,41,35,28)(22,45,47,55)(24,48,51,68)(25,49,26,54)(33,58,34,60)(38,61,63,72)(44,65,59,56)(46,66,67,71)(52,69,53,70);(3,8)(5,7)(9,13)(10,19)(11,24)(12,29)(14,33)(15,34)(16,32)(17,38)(18,40)(20,44)(21,43)(22,46)(23,48)(25,52)(26,53)(27,51)(28,56)(30,58)(31,57)(35,59)(36,60)(37,61)(39,63)(41,65)(42,64)(45,66)(47,67)(49,69)(50,68)(54,70)(55,71)(62,72);(2,6)(5,7)(13,32)(19,43)(24,51)(29,57)(33,34)(38,63)(40,64)(44,59)(46,67)(48,68)(52,53)(56,65)(58,60)(61,72)(66,71)(69,70)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410136);