Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #3455 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1358 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)(3,6)(4,5)(7,8)(9,10,13,19,14,20,28,40,15,18,29,38,30,39,50,60)(11,16,23,33,24,34,46,56,22,32,44,54,45,55,64,69)(12,17,26,36,27,37,49,59,25,35,47,57,48,58,65,70)(21,31,41,52,42,53,61,68,43,51,62,66,63,67,71,72);(2,5)(6,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,31)(18,32)(19,33)(20,34)(25,43)(26,41)(27,42)(28,46)(29,44)(30,45)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(47,62)(48,63)(49,61)(50,64)(57,66)(58,67)(59,68)(60,69)(65,71)(70,72);(9,12,13,26,14,27,28,49,15,25,29,47,30,48,50,65)(10,17,19,36,20,37,40,59,18,35,38,57,39,58,60,70)(11,21,23,41,24,42,46,61,22,43,44,62,45,63,64,71)(16,31,33,52,34,53,56,68,32,51,54,66,55,67,69,72);(1,3,4,7)(2,6,5,8)(11,22)(12,25)(16,32)(17,35)(23,44)(24,45)(26,47)(27,48)(33,54)(34,55)(36,57)(37,58)(46,64)(49,65)(56,69)(59,70);(10,18)(16,32)(17,35)(19,38)(20,39)(31,51)(33,54)(34,55)(36,57)(37,58)(40,60)(52,66)(53,67)(56,69)(59,70)(68,72)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,e^a=ei,d^b=di,d^c=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529858);