Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 |
| prime radical | #62879 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,14,24)(4,8,15,22)(5,17,19,9)(7,18,21,13)(12,25,26,32)(16,29,28,23)(20,30,31,27)(33,34,39,38)(35,40,46,45)(36,41,49,48)(37,42,50,51)(43,55,57,52)(44,53,58,59)(47,54,60,61)(56,63,64,62);(1,3,6,14)(2,7,10,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(9,24,17,11)(16,30,28,27)(23,32,29,25)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,52)(42,53)(47,56)(48,55)(49,57)(50,58)(51,59)(54,62)(60,64)(61,63);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32)(33,36)(34,41)(35,43)(37,47)(38,48)(39,49)(40,52)(42,54)(44,56)(45,55)(46,57)(50,60)(51,61)(53,62)(58,64)(59,63);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,37,39,50)(34,42,38,51)(35,44,46,58)(36,47,49,60)(40,53,45,59)(41,54,48,61)(43,56,57,64)(52,62,55,63);(33,38,39,34)(35,45,46,40)(36,48,49,41)(37,42,50,51)(43,55,57,52)(44,53,58,59)(47,54,60,61)(56,62,64,63)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259656);