Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,37,42)(35,44,47,39)(36,41,49,59)(38,52,51,43)(40,53,57,50)(45,55,46,58)(48,56,64,61)(54,62,60,63);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,25)(20,27)(22,28)(23,30)(33,35)(34,39)(36,45)(37,47)(38,48)(40,54)(41,58)(42,44)(43,56)(46,49)(50,62)(51,64)(52,61)(53,63)(55,59)(57,60);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,37,49)(34,40,42,57)(35,45,47,46)(38,53,51,50)(39,54,44,60)(41,43,59,52)(48,63,64,62)(55,61,58,56);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,25)(22,28)(26,31)(29,32)(33,36)(34,41)(35,46)(37,49)(38,50)(39,55)(40,52)(42,59)(43,57)(44,58)(45,47)(48,63)(51,53)(54,56)(60,61)(62,64);(2,8)(6,9)(7,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(34,42)(35,47)(36,49)(43,52)(48,64)(50,53)(54,60)(55,58)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7207566);