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 | |
3905 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,24,29)(19,30,32,25)(20,26,22,28)(21,27,31,23)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,24,32)(18,25,29,30)(20,31,22,21)(23,26,27,28)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,20,24,22)(18,26,29,28)(19,31,32,21)(23,25,27,30)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21)(18,27)(19,20)(22,32)(23,29)(24,31)(25,28)(26,30)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,22,24,20)(18,28,29,26)(19,31,32,21)(23,25,27,30)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,23,24,27)(18,21,29,31)(19,26,32,28)(20,30,22,25)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)Polycyclic presentation:
2^9|a^2=ghi,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, 8833803);