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 | |
3994 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(19,28)(21,32)(23,24)(26,29)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,48)(44,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,18,22,27)(19,24,28,23)(20,25,31,30)(21,26,32,29)(34,39)(36,41)(40,42)(43,47);(3,11)(4,12)(6,16)(8,13)(17,19,22,28)(18,23,27,24)(20,29,31,26)(21,25,32,30)(33,34,38,39)(35,40,45,42)(36,37,41,46)(43,44,47,48);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,27)(25,32)(29,31)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20)(18,25)(19,29)(21,24)(22,31)(23,32)(26,28)(27,30)(33,36,38,41)(34,37,39,46)(35,43,45,47)(40,44,42,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21)(18,26)(19,30)(20,24)(22,32)(23,31)(25,28)(27,29)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=g,e^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, 8834282);