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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,7)(5,9)(6,12)(10,14)(11,13)(15,16)(19,25)(23,30)(26,28)(31,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,45)(44,48)(46,47);(1,3)(2,8)(4,5)(6,11)(7,9)(10,15)(12,13)(14,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,46)(47,48);(2,4)(3,9)(6,14)(13,15)(18,22)(19,25)(24,27)(26,28)(34,38)(36,42)(41,46)(44,45);(1,4,7,2)(3,5,9,8)(6,12,14,10)(11,13,16,15)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,36,39,42)(34,40,38,35)(37,45,47,44)(41,43,46,48);(1,5)(2,9)(3,4)(6,13)(7,8)(10,16)(11,12)(14,15)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,47)(34,41,38,46)(35,43,40,48)(36,44,42,45);(1,6,7,14)(2,10,4,12)(3,11,9,16)(5,13,8,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38,39,34)(35,42,40,36)(37,46,47,41)(43,45,48,44)Polycyclic presentation:
2^9|d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837549);