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 |
| prime radical | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,10,14,15)(11,13,16,12)(19,25)(24,31)(26,29)(27,32)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,13,15,12)(17,18,22,23)(19,24,25,31)(20,28,30,21)(26,27,29,32)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(2,8)(3,9)(10,15)(11,16)(18,23)(21,28)(24,31)(27,32)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,4,6,7)(2,9,8,3)(5,13,14,12)(10,11,15,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,5,6,14)(2,10,8,15)(3,11,9,16)(4,12,7,13)(17,20)(18,21)(19,26)(22,30)(23,28)(24,32)(25,29)(27,31)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,16)(14,15)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=h,e^2=gh,f^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, 8833598);