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 | |
4035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,10,4,8)(6,9,14,13)(11,16,12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(35,42)(37,39)(43,47)(44,46);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(19,25)(22,29)(26,31)(30,32)(33,34)(35,37)(36,40)(38,41)(39,42)(43,46)(44,47)(45,48);(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,38,42)(34,39,41,37)(36,43,45,47)(40,44,48,46);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,46)(44,48)(45,47);(1,5,7,2)(3,8,4,10)(6,9,14,13)(11,16,12,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36,38,45)(34,40,41,48)(35,43,42,47)(37,46,39,44);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,37)(34,35)(36,44)(38,39)(40,47)(41,42)(43,48)(45,46)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818136);