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 | |
4133 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)(4,9)(5,10)(6,7)(11,16)(12,15)(13,14)(19,25)(23,30)(26,28)(31,32)(33,34)(35,37)(36,41)(38,40)(39,42)(43,45)(44,46)(47,48);(2,7)(4,14)(8,16)(10,12)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,38,42)(34,39,40,37)(36,43,46,48)(41,47,44,45);(3,11)(5,15)(8,16)(10,12)(18,22)(19,25)(24,27)(26,28)(34,40)(37,39)(41,44)(45,47);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,34)(35,39)(36,44)(37,42)(38,40)(41,46)(43,45)(47,48);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,44)(42,48);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37,38,39)(34,42,40,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|b^2=h,d^2=ghi,e^2=gh,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, 8837497);