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 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,11,8)(5,13,15,9)(10,16,14,12)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29)(33,34)(35,42)(36,46)(37,40)(38,47)(39,41)(43,48)(44,45);(2,7)(4,8)(9,13)(12,16)(17,19,23,20)(18,21,26,24)(22,27,30,28)(25,29,32,31)(33,35,39,37)(34,40,41,42)(36,44,47,43)(38,45,46,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,16)(12,13)(14,15)(17,19,23,20)(18,24,26,21)(22,27,30,28)(25,31,32,29)(33,36)(34,38)(35,43)(37,44)(39,47)(40,48)(41,46)(42,45);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,20,23,19)(18,24,26,21)(22,27,30,28)(25,29,32,31)(33,36)(34,38)(35,44)(37,43)(39,47)(40,45)(41,46)(42,48);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27)(33,37,39,35)(34,40,41,42)(36,44,47,43)(38,48,46,45);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,36)(35,45)(37,48)(39,46)(40,44)(41,47)(42,43)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327308);