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 | |
4190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,6)(5,13,15,10)(11,14,12,16)(20,28)(24,31)(27,32)(29,30)(33,34,39,43)(35,44,45,40)(36,41,46,48)(37,38,42,47);(2,8)(3,4)(10,13)(11,12)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(35,45)(36,46)(40,44)(41,48);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35)(34,40)(36,42)(37,46)(38,41)(39,45)(43,44)(47,48);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,46)(34,41,43,48)(35,42,45,37)(38,44,47,40);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,37,39,42)(34,38,43,47)(35,46,45,36)(40,41,44,48);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38)(34,42)(35,41)(36,40)(37,43)(39,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8342266);