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 | |
4142 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(19,25)(24,31)(26,29)(27,32)(33,34,39,41)(35,40,44,46)(36,45,43,38)(37,42,47,48);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,18,22,23)(19,24,25,31)(20,28,30,21)(26,27,29,32)(33,35)(34,40)(36,42)(37,38)(39,44)(41,46)(43,48)(45,47);(5,13)(8,15)(10,16)(12,14)(18,23)(21,28)(24,31)(27,32)(34,41)(35,44)(38,45)(42,48);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,36,39,43)(34,38,41,45)(35,42,44,48)(37,46,47,40);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,21)(19,26)(22,30)(23,28)(24,32)(25,29)(27,31)(33,37)(34,42)(35,38)(36,40)(39,47)(41,48)(43,46)(44,45);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,38)(34,43)(35,37)(36,41)(39,45)(40,48)(42,46)(44,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833510);