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 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,9,13,7)(4,8,14,5)(11,12,15,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34,39,41)(35,40,43,48)(36,44,45,42)(37,38,46,47);(1,3,6,13)(2,7,10,9)(4,11,14,15)(5,12,8,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(33,35)(34,40)(36,37)(38,44)(39,43)(41,48)(42,47)(45,46);(4,14)(5,8)(11,15)(12,16)(19,27)(20,28)(22,30)(24,32)(34,41)(35,43)(36,45)(38,47);(1,4)(2,8)(3,11)(5,10)(6,14)(7,16)(9,12)(13,15)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,36,39,45)(34,42,41,44)(35,37,43,46)(38,40,47,48);(1,3)(2,9)(4,15)(5,12)(6,13)(7,10)(8,16)(11,14)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,37)(34,38)(35,36)(39,46)(40,44)(41,47)(42,48)(43,45);(1,5,6,8)(2,4,10,14)(3,12,13,16)(7,11,9,15)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,38)(34,37)(35,44)(36,40)(39,47)(41,46)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=h,f^2=g,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, 8835972);