Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,4)(3,13,19,43)(5,23,26,51)(6,27,29,40)(7,10,31,20)(9,32,14,16)(11,36,21,25)(12,38,22,30)(15,45,49,63)(17,41,18,39)(24,37,56,52)(28,33,50,35)(34,59,44,55)(42,47,64,48)(46,53,58,60)(54,61,62,57);(1,3,7,18)(2,9,10,33)(4,14,20,35)(5,15,25,48)(6,28,30,16)(8,19,31,17)(11,34,23,53)(12,39,27,13)(21,44,51,60)(22,41,40,43)(24,55,57,46)(26,49,36,47)(29,50,38,32)(37,45,54,42)(52,63,62,64)(56,59,61,58);(2,10)(4,20)(6,29)(9,33)(11,23)(12,40)(13,41)(14,35)(16,32)(21,51)(22,27)(24,56)(28,50)(30,38)(34,53)(37,62)(39,43)(42,63)(44,60)(45,64)(46,58)(52,54)(55,59)(57,61);(1,4,8,2)(3,14,19,9)(5,11,26,21)(6,22,29,12)(7,20,31,10)(13,16,43,32)(15,34,49,44)(17,33,18,35)(23,36,51,25)(24,37,56,52)(27,30,40,38)(28,41,50,39)(42,58,64,46)(45,59,63,55)(47,60,48,53)(54,61,62,57);(1,5,8,26)(2,11,4,21)(3,15,19,49)(6,24,29,56)(7,25,31,36)(9,34,14,44)(10,23,20,51)(12,37,22,52)(13,42,43,64)(16,46,32,58)(17,47,18,48)(27,54,40,62)(28,55,50,59)(30,57,38,61)(33,53,35,60)(39,45,41,63)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259742);