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 | |
4111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,4,13)(5,16,10,6)(11,15,14,12)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(33,34,37,41)(35,39,43,38)(36,44,46,40)(42,47,45,48);(1,3)(2,8)(4,7)(5,11)(6,12)(9,13)(10,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35,37,43)(34,38,41,39)(36,42,46,45)(40,47,44,48);(2,9)(3,4)(6,16)(11,14)(19,27)(20,30)(23,31)(24,29)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(1,4,7,3)(2,8,9,13)(5,11,10,14)(6,15,16,12)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,5,7,10)(2,6,9,16)(3,11,4,14)(8,12,13,15)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,6)(2,10)(3,12)(4,15)(5,9)(7,16)(8,14)(11,13)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841382);