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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,10)(11,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,38,39)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,2)(3,10)(4,12)(5,8)(6,9)(7,13)(11,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(34,39)(40,42)(41,44)(47,48);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31)(33,35,38,37)(34,40,39,42)(36,45,46,43)(41,48,44,47);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36,38,46)(34,41,39,44)(35,43,37,45)(40,47,42,48);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664247);