Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,6)(3,7,4,8)(9,10,14,19)(11,21,24,16)(12,26,29,48)(13,31,32,49)(15,35,34,20)(17,37,44,30)(18,38,46,33)(22,51,53,42)(23,54,55,65)(25,41,36,47)(27,50,57,39)(28,43,62,68)(40,56,45,58)(52,70,72,66)(59,64,69,63)(60,67,61,71);(2,6)(7,8)(9,11,14,24)(10,16,19,21)(12,27,29,57)(13,23,32,55)(15,36,34,25)(17,42,44,51)(18,45,46,40)(20,47,35,41)(22,37,53,30)(26,39,48,50)(28,61,62,60)(31,65,49,54)(33,58,38,56)(43,67,68,71)(52,64,72,63)(59,66,69,70);(9,12)(10,17)(11,22)(13,28)(14,29)(15,37)(16,39)(18,43)(19,44)(20,48)(21,50)(23,52)(24,53)(25,57)(26,35)(27,36)(30,34)(31,59)(32,62)(33,64)(38,63)(40,66)(41,51)(42,47)(45,70)(46,68)(49,69)(54,71)(55,72)(56,60)(58,61)(65,67);(1,3,5,4)(2,7,6,8)(9,13,14,32)(10,18,19,46)(11,23,24,55)(12,28,29,62)(15,38,34,33)(16,40,21,45)(17,43,44,68)(20,49,35,31)(22,52,53,72)(25,58,36,56)(26,59,48,69)(27,60,57,61)(30,64,37,63)(39,66,50,70)(41,54,47,65)(42,67,51,71);(1,4,5,3)(2,7,6,8)(10,19)(11,24)(17,44)(18,46)(20,35)(22,53)(23,55)(25,36)(26,48)(27,57)(31,49)(43,68)(52,72)(56,58)(59,69)(60,61)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429084);