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 | |
1289 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)(9,10,13,19,14,20,29,41,15,17,30,36,31,37,53,60)(11,16,23,34,24,35,48,59,21,32,43,54,44,55,64,69)(12,25,26,49,27,50,51,68,28,18,38,39,52,40,61,62)(22,45,46,65,47,66,67,72,42,33,56,57,63,58,70,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,32)(18,33)(19,34)(20,35)(25,45)(26,46)(27,47)(28,42)(29,48)(30,43)(31,44)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(49,65)(50,66)(51,67)(52,63)(53,64)(60,69)(61,70)(62,71)(68,72);(3,8)(5,7)(10,17)(11,21)(18,25)(19,36)(20,37)(22,42)(23,43)(24,44)(39,49)(40,50)(41,60)(46,56)(47,63)(48,64)(62,68)(67,70);(2,6)(5,7)(11,21)(16,32)(22,42)(23,43)(24,44)(33,45)(34,54)(35,55)(46,56)(47,63)(48,64)(57,65)(58,66)(59,69)(67,70)(71,72);(9,12,13,26,14,27,29,51,15,28,30,38,31,52,53,61)(10,18,19,39,20,40,41,62,17,25,36,49,37,50,60,68)(11,22,23,46,24,47,48,67,21,42,43,56,44,63,64,70)(16,33,34,57,35,58,59,71,32,45,54,65,55,66,69,72)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,e^a=ei,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, 7529976);