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 | |
4081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,5)(6,9)(8,12)(10,11)(13,16)(14,15)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34)(35,43)(36,45)(37,42)(38,41)(39,40)(44,47)(46,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,14)(15,16)(17,19)(18,22)(20,29)(21,30)(23,26)(24,25)(27,28)(31,32)(33,35)(34,37)(36,44)(38,48)(39,43)(40,42)(41,47)(45,46);(1,3,6,12)(2,7,9,8)(4,10,14,13)(5,11,15,16)(17,20,23,30)(18,24,27,31)(19,29,26,21)(22,25,28,32)(33,36,40,38)(34,41,39,45)(35,44,42,48)(37,47,43,46);(1,3)(2,8)(4,13)(5,11)(6,12)(7,9)(10,14)(15,16)(17,20)(18,24)(19,29)(21,26)(22,25)(23,30)(27,31)(28,32)(33,37,40,43)(34,42,39,35)(36,46,38,47)(41,44,45,48);(1,4,6,14)(2,5,9,15)(3,10,12,13)(7,11,8,16)(17,21,23,29)(18,25,27,32)(19,30,26,20)(22,24,28,31)(33,38,40,36)(34,41,39,45)(35,44,42,48)(37,46,43,47);(1,5)(2,4)(3,11)(6,15)(7,10)(8,13)(9,14)(12,16)(17,22,23,28)(18,26,27,19)(20,25,30,32)(21,31,29,24)(33,39,40,34)(35,43,42,37)(36,45,38,41)(44,46,48,47)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113956);