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 | #46052 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,9,5,7)(4,8,12,15)(11,16,13,14)(20,28)(24,29)(26,32)(30,31)(33,34,38,39)(35,36,43,40)(37,41,45,46)(42,44,48,47);(1,3,6,5)(2,7,10,9)(4,11,12,13)(8,14,15,16)(18,22)(23,25)(24,29)(30,31)(34,39)(36,40)(41,46)(44,47);(4,12)(8,15)(11,13)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(33,36,38,40)(34,35,39,43)(37,44,45,47)(41,42,46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5,6,3)(2,9,10,7)(4,13,12,11)(8,16,15,14)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,41,39,46)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002221);