Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C24 |
| commutator, socle | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | #3529 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1291 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)(3,7)(4,6)(5,8)(9,10,13,19,14,20,28,40,15,17,29,36,30,37,50,59)(11,16,23,33,24,34,46,56,21,31,42,52,43,53,63,68)(12,18,25,38,26,39,47,60,27,35,48,57,49,58,65,70)(22,32,44,54,45,55,64,69,41,51,61,66,62,67,71,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,31)(18,32)(19,33)(20,34)(25,44)(26,45)(27,41)(28,46)(29,42)(30,43)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(47,64)(48,61)(49,62)(50,63)(57,66)(58,67)(59,68)(60,69)(65,71)(70,72);(3,8)(5,7)(10,17)(11,21)(18,35)(19,36)(20,37)(22,41)(23,42)(24,43)(38,57)(39,58)(40,59)(44,61)(45,62)(46,63)(60,70)(64,71);(2,6)(5,7)(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72);(9,12,13,25,14,26,28,47,15,27,29,48,30,49,50,65)(10,18,19,38,20,39,40,60,17,35,36,57,37,58,59,70)(11,22,23,44,24,45,46,64,21,41,42,61,43,62,63,71)(16,32,33,54,34,55,56,69,31,51,52,66,53,67,68,72)Polycyclic presentation:
2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,d^b=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529955);