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 | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,6,4,9)(5,13,15,10)(11,14,12,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,31)(30,32)(33,34,39,44)(35,45,47,40)(36,38,43,41)(37,48,46,42);(1,3)(2,6)(4,7)(5,11)(8,9)(10,16)(12,15)(13,14)(17,19,22,24)(18,21,25,26)(20,27,29,32)(23,31,28,30)(33,35)(34,40)(36,37)(38,42)(39,47)(41,48)(43,46)(44,45);(2,8)(3,4)(10,13)(11,12)(20,29)(23,28)(27,32)(30,31)(33,36,39,43)(34,41,44,38)(35,37,47,46)(40,48,45,42);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,14,13,16)(17,20)(18,23)(19,27)(21,31)(22,29)(24,32)(25,28)(26,30)(33,36)(34,41)(35,46)(37,47)(38,44)(39,43)(40,42)(45,48);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,16,9,14)(17,18)(19,21)(20,23)(22,25)(24,26)(27,31)(28,29)(30,32)(33,37,39,46)(34,42,44,48)(35,36,47,43)(38,45,41,40);(1,6,7,9)(2,4,8,3)(5,14,15,16)(10,11,13,12)(17,21)(18,24)(19,25)(20,30)(22,26)(23,27)(28,32)(29,31)(33,38,39,41)(34,43,44,36)(35,42,47,48)(37,40,46,45)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627709);