Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563327 |
| 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 | C23 | C2×H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,25)(13,31)(14,34)(15,20)(17,29)(18,33)(19,27)(21,40)(22,49)(23,45)(24,39)(26,36)(28,57)(30,53)(32,60)(35,38)(37,51)(41,58)(42,46)(43,61)(44,56)(47,67)(48,65)(50,68)(52,59)(54,64)(55,63)(62,70)(66,72)(69,71);(1,3,5,4)(2,6,8,7)(9,11,15,24)(10,16,20,39)(12,26,30,55)(13,22,32,50)(14,35,27,23)(17,40,42,65)(18,43,44,37)(19,45,34,38)(21,46,48,29)(25,36,53,63)(28,54,58,72)(31,49,60,68)(33,61,56,51)(41,66,57,64)(47,69,70,59)(52,67,71,62);(9,12,27,46,15,30,14,29)(10,17,34,53,20,42,19,25)(11,21,35,55,24,48,23,26)(13,28,56,69,32,58,33,59)(16,36,45,65,39,63,38,40)(18,41,60,71,44,57,31,52)(22,47,61,72,50,70,51,54)(37,62,68,66,43,67,49,64);(9,13,15,32)(10,18)(11,22,24,50)(12,28,30,58)(14,33,27,56)(16,37)(17,41)(19,31)(20,44)(21,47,48,70)(23,51,35,61)(25,52)(26,54,55,72)(29,59,46,69)(34,60)(36,62)(38,49)(39,43)(40,64)(42,57)(45,68)(53,71)(63,67)(65,66);(1,4,5,3)(2,7,8,6)(13,32)(18,44)(22,50)(28,58)(31,60)(33,56)(37,43)(41,57)(47,70)(49,68)(51,61)(52,71)(54,72)(59,69)(62,67)(64,66)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521320);