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 | |
4122 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)(3,11)(4,13)(5,10)(6,7)(8,14)(9,15)(12,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34,39,41)(35,40,43,37)(36,42,45,47)(38,44,48,46);(2,7)(4,15)(8,11)(10,12)(17,19,22,24)(18,23,26,20)(21,27,29,31)(25,30,32,28)(33,35)(34,40)(36,38)(37,41)(39,43)(42,44)(45,48)(46,47);(2,7)(5,16)(8,11)(9,13)(21,29)(25,32)(27,31)(28,30)(34,41)(35,43)(38,48)(42,47);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,36)(34,42)(35,38)(37,46)(39,45)(40,44)(41,47)(43,48);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,20)(18,24)(19,26)(21,28)(22,23)(25,31)(27,32)(29,30)(33,37,39,40)(34,43,41,35)(36,44,45,46)(38,47,48,42);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,38)(34,44)(35,36)(37,47)(39,48)(40,42)(41,46)(43,45)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8300945);