Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1278 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)(11,17,24,39)(12,27,30,56)(13,19,15,36)(14,34,35,61)(18,41,43,31)(20,33,44,28)(22,45,50,68)(23,38,26,54)(25,53,52,72)(29,55,32,60)(37,62,64,48)(40,51,65,46)(42,58,57,59)(47,66,49,71)(63,69,67,70);(1,3)(2,5)(4,8)(6,7)(9,11,16,24)(10,17,21,39)(12,22,30,50)(13,23,15,26)(14,25,35,52)(18,37,43,64)(19,38,36,54)(20,40,44,65)(27,45,56,68)(28,46,33,51)(29,47,32,49)(31,48,41,62)(34,53,61,72)(42,63,57,67)(55,66,60,71)(58,69,59,70);(9,12,28,58,15,32,35,41,16,30,33,59,13,29,14,31)(10,18,34,55,19,42,44,56,21,43,61,60,36,57,20,27)(11,22,46,69,26,49,52,62,24,50,51,70,23,47,25,48)(17,37,53,66,38,63,65,68,39,64,72,71,54,67,40,45);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,29,30,32)(14,33,35,28)(17,38,39,54)(18,42,43,57)(20,34,44,61)(22,47,50,49)(25,51,52,46)(27,55,56,60)(31,59,41,58)(37,63,64,67)(40,53,65,72)(45,66,68,71)(48,70,62,69);(2,6)(5,7)(11,24)(12,30)(17,39)(18,43)(23,26)(25,52)(27,56)(29,32)(31,41)(38,54)(40,65)(42,57)(46,51)(53,72)(55,60)(58,59)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532103);