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 | #54156 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(4,12)(7,10)(9,15)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(36,44)(40,45)(42,48)(46,47);(1,2,6,5)(3,7,11,10)(4,8,12,13)(9,14,15,16)(18,20)(19,25)(24,28)(27,31)(34,38)(39,41)(40,45)(46,47);(1,3,6,11)(2,7,5,10)(4,9,12,15)(8,14,13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4)(2,8)(3,9)(5,13)(6,12)(7,14)(10,16)(11,15)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,2)(3,10,11,7)(4,13,12,8)(9,16,15,14)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=i,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, 8026845);