Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| 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 | #24293 | 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)(9,10,16,21)(11,17,25,42)(12,26,29,46)(13,19,33,50)(14,34,15,20)(18,36,45,30)(22,43,52,67)(23,39,55,66)(24,41,35,40)(27,56,53,38)(28,57,60,71)(31,47,54,65)(32,49,37,48)(44,62,70,61)(51,68,59,72)(58,69,63,64);(2,5)(6,8)(9,11)(10,17)(12,27)(13,31)(14,24)(15,35)(16,25)(18,43)(19,47)(20,40)(21,42)(22,30)(23,37)(26,56)(28,59)(29,53)(32,55)(33,54)(34,41)(36,52)(38,46)(39,48)(44,69)(45,67)(49,66)(50,65)(51,60)(57,72)(58,61)(62,63)(64,70)(68,71);(9,12,15,36,16,29,14,30)(10,18,34,46,21,45,20,26)(11,22,35,27,25,52,24,53)(13,28,37,62,33,60,32,61)(17,38,41,67,42,56,40,43)(19,44,49,71,50,70,48,57)(23,51,54,58,55,59,31,63)(39,64,47,72,66,69,65,68);(9,13)(10,19)(11,23)(12,28)(14,32)(15,37)(16,33)(17,39)(18,44)(20,48)(21,50)(22,51)(24,31)(25,55)(26,57)(27,58)(29,60)(30,61)(34,49)(35,54)(36,62)(38,64)(40,65)(41,47)(42,66)(43,68)(45,70)(46,71)(52,59)(53,63)(56,69)(67,72);(1,3,4,7)(2,6,5,8)(12,29)(18,45)(22,52)(26,46)(27,53)(28,60)(30,36)(38,56)(43,67)(44,70)(51,59)(57,71)(58,63)(61,62)(64,69)(68,72)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,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, 6352411);