Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2668 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,21)(5,11)(6,12)(7,13)(8,14)(9,19)(10,23)(16,34)(17,43)(18,33)(20,32)(22,51)(24,36)(25,56)(26,39)(27,40)(28,41)(29,37)(30,47)(31,48)(35,54)(38,49)(42,46)(44,57)(45,58)(50,55)(52,60)(53,61)(59,63)(62,64);(1,3)(2,9)(4,16)(5,25)(6,18)(7,19)(8,20)(10,34)(11,38)(12,15)(13,32)(14,33)(17,27)(21,42)(22,52)(23,45)(24,47)(26,49)(28,48)(29,46)(30,37)(31,39)(35,60)(36,58)(40,56)(41,43)(44,54)(50,57)(51,59)(53,63)(55,64)(61,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,36)(13,21)(14,37)(15,42)(17,44)(18,45)(19,46)(20,47)(25,52)(26,53)(27,54)(28,55)(31,57)(32,58)(33,34)(38,62)(39,51)(40,50)(41,61)(43,60)(48,64)(49,63)(56,59);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,31)(10,35)(12,39)(13,40)(14,41)(15,43)(16,44)(18,25)(19,48)(20,49)(21,50)(23,53)(24,55)(29,54)(30,57)(32,38)(33,56)(34,59)(36,51)(37,61)(42,60)(45,52)(46,64)(47,63)(58,62);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,23,24,29)(5,26,28,27)(9,15,33,32)(10,36,37,21)(11,39,41,40)(16,45,47,46)(17,25,49,48)(22,53,55,54)(30,42,34,58)(31,43,56,38)(35,51,61,50)(44,52,63,64)(57,60,59,62)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407950);