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 | |
4079 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)(8,10)(9,15)(11,14)(12,16)(18,19)(23,25)(24,26)(31,32)(33,34)(35,40)(36,38)(37,42)(39,41)(43,45)(44,47)(46,48);(2,7)(4,8)(9,13)(12,14)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,35,39,36)(34,40,41,38)(37,44,46,43)(42,47,48,45);(2,7)(3,10)(9,13)(11,16)(34,41)(35,36)(42,48)(43,44);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,36,39,35)(34,38,41,40)(37,43,46,44)(42,45,48,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,46)(34,42,41,48)(35,43,36,44)(38,47,40,45);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,38)(34,35)(36,41)(37,45)(39,40)(42,44)(43,48)(46,47)Polycyclic presentation:
2^9|b^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073656);