Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814827 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10,14,19)(11,22,25,17)(12,27,29,59)(13,20,35,47)(15,38,16,21)(18,32,46,31)(23,44,53,68)(24,52,56,41)(26,42,39,43)(28,51,60,40)(30,58,34,65)(33,64,57,48)(36,66,37,50)(45,63,49,62)(54,70,55,71)(61,69,72,67);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,33)(14,25)(15,26)(16,39)(18,44)(19,22)(20,48)(21,42)(23,32)(24,36)(27,40)(29,60)(30,54)(31,53)(34,55)(35,57)(37,56)(38,43)(41,66)(45,67)(46,68)(47,64)(49,69)(50,52)(51,59)(58,71)(61,62)(63,72)(65,70);(2,5)(6,8)(9,12,14,29)(10,18)(11,23)(13,34,35,30)(15,31,16,32)(17,40,22,51)(19,46)(20,49)(21,27)(24,55)(25,53)(26,28)(33,61)(36,63,37,62)(38,59)(39,60)(41,69,52,67)(42,68,43,44)(45,47)(48,70,64,71)(50,65)(54,56)(57,72)(58,66);(1,3,4,7)(2,6,5,8)(10,19)(12,29)(17,22)(20,47)(21,38)(23,53)(28,60)(30,34)(31,32)(41,52)(42,43)(48,64)(50,66)(54,55)(61,72)(62,63);(9,13,15,36,14,35,16,37)(10,20,38,66,19,47,21,50)(11,24,39,33,25,56,26,57)(12,30,31,62,29,34,32,63)(17,41,42,48,22,52,43,64)(18,45,59,65,46,49,27,58)(23,54,60,72,53,55,28,61)(40,67,68,71,51,69,44,70)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=fi,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724438);