Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636940 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1114 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34,38,41)(35,39,44,54)(36,46,49,63)(37,51,48,42)(40,55,57,50)(43,58,47,62)(45,61,59,53)(52,64,56,60);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30)(33,35,38,44)(34,39,41,54)(36,47,49,43)(37,45,48,59)(40,56,57,52)(42,53,51,61)(46,62,63,58)(50,64,55,60);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(19,25)(22,29)(26,27)(30,31)(33,36,48,55,38,49,37,50)(34,40,51,63,41,57,42,46)(35,43,59,64,44,47,45,60)(39,52,61,62,54,56,53,58);(2,9)(3,12)(8,14)(11,13)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,35)(34,39)(36,47)(37,45)(38,44)(40,56)(41,54)(42,53)(43,49)(46,62)(48,59)(50,64)(51,61)(52,57)(55,60)(58,63);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,24)(22,29)(23,32)(30,31)(34,41)(35,44)(36,49)(42,51)(45,59)(50,55)(52,56)(58,62)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fgi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6557470);