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 | #53062 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,8)(9,14)(10,15)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,34)(35,39)(36,43)(37,42)(38,48)(40,45)(41,47)(44,46);(1,2,6,7)(3,4,8,12)(5,9,13,14)(10,11,15,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,41)(36,46)(37,38)(39,45)(40,47)(42,44)(43,48);(5,13)(9,14)(10,15)(11,16)(19,22)(24,26)(27,29)(31,32)(33,36,40,37)(34,42,45,43)(35,46,47,38)(39,48,41,44);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,15)(14,16)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,37,40,36)(34,43,45,42)(35,46,47,38)(39,48,41,44);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,38,40,46)(34,44,45,48)(35,37,47,36)(39,43,41,42);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,11,12,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,39,40,41)(34,35,45,47)(36,44,37,48)(38,43,46,42)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8253129);