Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| 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 | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 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)(11,22)(12,27)(13,19)(14,20)(15,35)(16,21)(17,25)(18,31)(23,53)(24,52)(26,41)(28,39)(29,59)(30,46)(32,64)(33,48)(34,50)(36,42)(37,45)(38,49)(40,57)(43,55)(44,62)(47,58)(51,56)(54,70)(60,65)(61,68)(63,67)(66,71)(69,72);(1,3,5,4)(2,6,8,7)(9,11,14,25)(10,17,20,22)(12,28,30,56)(13,32,33,58)(15,26,16,36)(18,43,45,53)(19,47,48,64)(21,41,35,42)(23,37,55,31)(24,38,57,34)(27,51,46,39)(29,54,61,71)(40,49,52,50)(44,65,67,69)(59,66,68,70)(60,62,72,63);(1,3,5,4)(2,7,8,6)(9,12,15,31,14,30,16,37)(10,18,21,27,20,45,35,46)(11,23,26,56,25,55,36,28)(13,29,34,62,33,61,38,63)(17,39,41,43,22,51,42,53)(19,44,49,59,48,67,50,68)(24,54,58,72,57,71,32,60)(40,65,47,66,52,69,64,70);(1,4,5,3)(2,6,8,7)(9,13,14,33)(10,19,20,48)(11,24,25,57)(12,29,30,61)(15,34,16,38)(17,40,22,52)(18,44,45,67)(21,49,35,50)(23,54,55,71)(26,58,36,32)(27,59,46,68)(28,60,56,72)(31,62,37,63)(39,65,51,69)(41,47,42,64)(43,66,53,70);(11,25)(17,22)(23,55)(24,57)(26,36)(28,56)(32,58)(39,51)(40,52)(41,42)(43,53)(47,64)(54,71)(60,72)(65,69)(66,70)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351924);