Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24290 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1917 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)(3,6)(4,5)(7,8)(9,10,14,20)(11,22,25,17)(12,27,30,46)(13,19,33,48)(15,35,16,21)(18,37,45,31)(23,53,55,43)(24,52,57,40)(26,41,36,42)(28,39,56,51)(29,59,61,68)(32,64,58,47)(34,50,38,49)(44,63,67,62)(54,70,71,66)(60,65,72,69);(2,5)(6,8)(9,11)(10,17)(12,28)(13,32)(14,25)(15,26)(16,36)(18,43)(19,47)(20,22)(21,41)(23,31)(24,34)(27,51)(29,54)(30,56)(33,58)(35,42)(37,55)(38,57)(39,46)(40,50)(44,65)(45,53)(48,64)(49,52)(59,66)(60,63)(61,71)(62,72)(67,69)(68,70);(9,12,16,37,14,30,15,31)(10,18,35,46,20,45,21,27)(11,23,36,28,25,55,26,56)(13,29,38,63,33,61,34,62)(17,39,42,53,22,51,41,43)(19,44,50,68,48,67,49,59)(24,54,32,60,57,71,58,72)(40,65,64,70,52,69,47,66);(9,13)(10,19)(11,24)(12,29)(14,33)(15,34)(16,38)(17,40)(18,44)(20,48)(21,49)(22,52)(23,54)(25,57)(26,58)(27,59)(28,60)(30,61)(31,62)(32,36)(35,50)(37,63)(39,65)(41,47)(42,64)(43,66)(45,67)(46,68)(51,69)(53,70)(55,71)(56,72);(1,3,4,7)(2,6,5,8)(11,25)(17,22)(23,55)(24,57)(26,36)(28,56)(32,58)(39,51)(40,52)(41,42)(43,53)(47,64)(54,71)(60,72)(65,69)(66,70)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352390);