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 | |
4168 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)(3,8,4,11)(5,9,13,14)(10,15,12,16)(17,18,22,23)(19,20,27,24)(21,29,31,25)(26,32,30,28)(33,34)(35,42)(36,46)(37,40)(38,47)(39,41)(43,48)(44,45);(2,7)(8,11)(9,14)(15,16)(18,23)(20,24)(25,29)(28,32)(33,35,39,37)(34,40,41,42)(36,44,47,43)(38,45,46,48);(1,3,6,4)(2,8,7,11)(5,10,13,12)(9,15,14,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36)(34,38)(35,43)(37,44)(39,47)(40,48)(41,46)(42,45);(1,4,6,3)(2,8,7,11)(5,10,13,12)(9,16,14,15)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32)(33,36)(34,38)(35,44)(37,43)(39,47)(40,45)(41,46)(42,48);(1,2)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,37,39,35)(34,40,41,42)(36,44,47,43)(38,48,46,45);(1,5,6,13)(2,9,7,14)(3,10,4,12)(8,15,11,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)(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=i,c^2=gh,d^2=gh,e^2=h,f^2=gh,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, 10329906);