Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(34,40)(36,38)(41,48)(44,46);(2,7)(3,9)(10,13)(11,16)(18,20)(19,25)(24,28)(27,31)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,46)(47,48);(2,7)(4,8)(10,13)(12,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,42)(34,36,40,38)(37,45,47,43)(41,46,48,44);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36,39,38)(34,35,40,42)(37,44,47,46)(41,43,48,45);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,47)(34,41,40,48)(35,43,42,45)(36,44,38,46);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,36)(34,42,40,35)(37,46,47,44)(41,45,48,43)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026775);