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 | |
4277 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)(3,8,4,10)(6,13,14,9)(11,16,12,15)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32)(33,34,38,40)(35,36,41,43)(37,39,45,46)(42,44,47,48);(2,5)(3,4)(6,14)(15,16)(18,23)(20,26)(21,30)(27,31)(35,41)(36,43)(37,45)(39,46);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,35,38,41)(34,36,40,43)(37,42,45,47)(39,44,46,48);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32)(33,36)(34,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)(33,37,38,45)(34,39,40,46)(35,42,41,47)(36,44,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^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, 8961701);