Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500755 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2661 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)(9,18)(10,23)(16,34)(17,43)(19,33)(20,32)(22,51)(24,36)(25,56)(26,39)(27,40)(28,41)(29,37)(30,46)(31,47)(35,54)(38,48)(42,45)(44,57)(49,58)(50,55)(52,60)(53,61)(59,64)(62,63);(1,3,8,20)(2,9,14,33)(4,16,24,46)(5,25,28,47)(6,19,7,18)(10,34,37,30)(11,38,41,43)(12,15,13,32)(17,26,48,27)(21,42,36,58)(22,52,55,63)(23,49,29,45)(31,40,56,39)(35,60,61,62)(44,53,64,54)(50,57,51,59);(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)(19,49)(20,46)(25,52)(26,53)(27,54)(28,55)(31,57)(32,58)(33,34)(38,62)(39,51)(40,50)(41,61)(43,60)(47,63)(48,64)(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)(18,47)(19,25)(20,48)(21,50)(23,53)(24,55)(29,54)(30,57)(32,38)(33,56)(34,59)(36,51)(37,61)(42,60)(45,63)(46,64)(49,52)(58,62);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,23,24,29)(5,26,28,27)(9,32,33,15)(10,36,37,21)(11,39,41,40)(16,45,46,49)(17,47,48,25)(22,53,55,54)(30,58,34,42)(31,38,56,43)(35,51,61,50)(44,63,64,52)(57,62,59,60)Polycyclic presentation:
2^9|b^2=i,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,e^b=ei,f^b=fi,g^b=gi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407954);