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 | |
4156 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,34)(35,40)(36,46)(37,41)(38,47)(39,42)(43,44)(45,48);(1,3,6,8)(2,4,7,10)(5,11,14,15)(9,12,13,16)(17,18,21,24)(19,22,27,25)(20,23,29,32)(26,30,28,31)(33,35,39,45)(34,40,42,48)(36,41,47,43)(37,38,44,46);(2,7)(4,10)(5,14)(11,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30)(33,36,39,47)(34,38,42,46)(35,43,45,41)(37,48,44,40);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36)(34,38)(35,41)(37,48)(39,47)(40,44)(42,46)(43,45);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37,39,44)(34,41,42,43)(35,38,45,46)(36,40,47,48);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,38)(34,36)(35,44)(37,45)(39,46)(40,41)(42,47)(43,48)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961131);