Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #62276 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1271 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)(9,10,16,21)(11,17,25,42)(12,26,29,55)(13,19,15,37)(14,35,36,64)(18,43,45,31)(20,34,47,28)(22,48,27,54)(23,39,33,41)(24,52,53,67)(30,56,32,63)(38,60,44,51)(40,58,68,49)(46,61,57,62)(50,69,59,70)(65,72,66,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,27)(13,33)(14,24)(15,23)(16,25)(18,44)(19,41)(20,40)(21,42)(22,29)(26,54)(28,49)(30,50)(31,60)(32,59)(34,58)(35,52)(36,53)(37,39)(38,45)(43,51)(46,65)(47,68)(48,55)(56,69)(57,66)(61,72)(62,71)(63,70)(64,67);(9,12,28,61,15,32,36,43,16,29,34,62,13,30,14,31)(10,18,35,56,19,46,47,55,21,45,64,63,37,57,20,26)(11,22,49,71,23,50,53,60,25,27,58,72,33,59,24,51)(17,38,52,70,41,66,68,54,42,44,67,69,39,65,40,48);(3,8)(5,7)(12,29)(18,45)(22,27)(26,55)(30,32)(31,43)(38,44)(46,57)(48,54)(50,59)(51,60)(56,63)(61,62)(65,66)(69,70)(71,72);(2,6)(5,7)(9,13,16,15)(10,19,21,37)(11,23,25,33)(12,30,29,32)(14,34,36,28)(17,39,42,41)(18,46,45,57)(20,35,47,64)(22,50,27,59)(24,49,53,58)(26,56,55,63)(31,62,43,61)(38,65,44,66)(40,67,68,52)(48,69,54,70)(51,71,60,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532048);