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 | #38308 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,15)(11,12)(14,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,37)(35,41,46,40)(36,43,38,42)(44,47,45,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,46)(42,47)(43,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35)(34,41)(36,44)(37,40)(38,45)(39,46)(42,48)(43,47);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36,39,38)(34,42,37,43)(35,44,46,45)(40,47,41,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,34)(35,41,46,40)(36,43,38,42)(44,48,45,47);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,36)(34,43,37,42)(35,45,46,44)(40,48,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026468);