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 | |
1356 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)(9,10,13,18,14,19,28,38,15,20,29,39,30,40,50,60)(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)(21,31,42,52,43,53,63,68,41,51,61,66,62,67,71,72);(2,5)(6,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,31)(18,33)(19,34)(20,32)(25,41)(26,42)(27,43)(28,46)(29,44)(30,45)(35,51)(36,52)(37,53)(38,56)(39,54)(40,55)(47,61)(48,62)(49,63)(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)(10,17,18,36,19,37,38,59,20,35,39,57,40,58,60,70)(11,21,23,42,24,43,46,63,22,41,44,61,45,62,64,71)(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)(16,32)(21,41)(23,44)(24,45)(31,51)(33,54)(34,55)(42,61)(43,62)(46,64)(52,66)(53,67)(56,69)(63,71)(68,72);(12,25)(17,35)(21,41)(26,47)(27,48)(31,51)(36,57)(37,58)(42,61)(43,62)(49,65)(52,66)(53,67)(59,70)(63,71)(68,72)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,d^b=di,e^c=ei176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529859);