Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350909 |
| 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 | C42 | C25 |
| prime radical | #62938 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,10)(3,12,15,23)(4,9,16,25)(5,11,6,19)(8,20,22,14)(13,26,27,31)(17,24,18,29)(21,30,32,28)(33,34)(35,46)(36,51)(37,43)(38,44)(39,45)(40,49)(41,52)(42,54)(47,60)(48,58)(50,57)(53,59)(55,63)(56,62)(61,64);(1,3,7,15)(2,8,10,22)(4,13,16,27)(5,14,6,20)(9,21,25,32)(11,23,19,12)(17,28,18,30)(24,31,29,26)(33,35,39,50)(34,40,45,58)(36,47,54,63)(37,48,38,49)(41,55,59,60)(42,56,51,64)(43,46,44,57)(52,61,53,62);(1,4,7,16)(2,9,10,25)(3,13,15,27)(5,17,6,18)(8,21,22,32)(11,24,19,29)(12,26,23,31)(14,28,20,30)(33,36,39,54)(34,41,45,59)(35,47,50,63)(37,52,38,53)(40,55,58,60)(42,44,51,43)(46,56,57,64)(48,61,49,62);(2,10)(4,16)(8,22)(11,19)(12,23)(13,27)(17,18)(28,30)(33,36)(34,42)(35,47)(37,53)(38,52)(39,54)(40,56)(41,43)(44,59)(45,51)(46,55)(48,62)(49,61)(50,63)(57,60)(58,64);(1,5,7,6)(2,11,10,19)(3,14,15,20)(4,17,16,18)(8,23,22,12)(9,24,25,29)(13,28,27,30)(21,31,32,26)(33,37,39,38)(34,43,45,44)(35,48,50,49)(36,52,54,53)(40,46,58,57)(41,42,59,51)(47,61,63,62)(55,56,60,64)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258152);