Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| 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 | #3529 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1287 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)(9,10,13,18,14,19,29,40,15,20,30,41,31,42,53,64)(11,16,23,34,24,35,47,58,21,32,43,54,44,55,65,69)(12,17,27,38,28,39,52,63,26,37,50,61,51,62,68,72)(22,33,45,56,46,57,66,70,25,36,48,59,49,60,67,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,23)(14,24)(15,21)(17,36)(18,34)(19,35)(20,32)(22,26)(27,48)(28,49)(29,47)(30,43)(31,44)(33,37)(38,59)(39,60)(40,58)(41,54)(42,55)(45,50)(46,51)(52,67)(53,65)(56,61)(57,62)(63,71)(64,69)(66,68)(70,72);(3,8)(5,7)(11,21)(12,26)(16,32)(17,37)(23,43)(24,44)(27,50)(28,51)(34,54)(35,55)(38,61)(39,62)(47,65)(52,68)(58,69)(63,72);(2,6)(5,7)(11,21)(16,32)(22,25)(23,43)(24,44)(33,36)(34,54)(35,55)(45,48)(46,49)(47,65)(56,59)(57,60)(58,69)(66,67)(70,71);(9,12,13,27,14,28,29,52,15,26,30,50,31,51,53,68)(10,17,18,38,19,39,40,63,20,37,41,61,42,62,64,72)(11,22,23,45,24,46,47,66,21,25,43,48,44,49,65,67)(16,33,34,56,35,57,58,70,32,36,54,59,55,60,69,71)Polycyclic presentation:
2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cfi,d^b=di,e^b=ei,e^c=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529959);