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 | #62275 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1273 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)(11,17,25,41)(12,26,28,53)(13,19,15,36)(14,34,35,61)(18,42,43,30)(20,33,45,27)(22,46,48,68)(23,38,32,40)(24,51,52,66)(29,54,31,59)(37,62,63,50)(39,60,67,47)(44,56,55,58)(49,69,57,70)(64,72,65,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,32)(14,24)(15,23)(16,25)(18,37)(19,40)(20,39)(21,41)(26,46)(27,47)(28,48)(29,57)(30,50)(31,49)(33,60)(34,51)(35,52)(36,38)(42,62)(43,63)(44,65)(45,67)(53,68)(54,70)(55,64)(56,71)(58,72)(59,69)(61,66);(9,12,27,56,15,31,35,42,16,28,33,58,13,29,14,30)(10,18,34,54,19,44,45,53,21,43,61,59,36,55,20,26)(11,22,47,71,23,49,52,62,25,48,60,72,32,57,24,50)(17,37,51,70,40,65,67,68,41,63,66,69,38,64,39,46);(3,8)(5,7)(12,28)(18,43)(22,48)(26,53)(29,31)(30,42)(37,63)(44,55)(46,68)(49,57)(50,62)(54,59)(56,58)(64,65)(69,70)(71,72);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,23,25,32)(12,29,28,31)(14,33,35,27)(17,38,41,40)(18,44,43,55)(20,34,45,61)(22,49,48,57)(24,47,52,60)(26,54,53,59)(30,58,42,56)(37,64,63,65)(39,66,67,51)(46,69,68,70)(50,71,62,72)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,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, 7532042);