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 |
| prime radical | #57902 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,15)(11,14)(12,16)(17,18)(19,28)(20,25)(21,31)(22,29)(23,27)(24,30)(26,32)(33,34)(35,41)(36,45)(37,47)(38,40)(39,42)(43,44)(46,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,19)(18,24)(20,22)(21,29)(23,30)(25,26)(27,28)(31,32)(33,34,39,42)(35,40,38,41)(36,45,47,37)(43,48,46,44);(1,3,6,7)(2,4,9,10)(5,11,13,14)(8,12,15,16)(17,20,23,31)(18,25,27,21)(19,22,30,32)(24,26,28,29)(33,35,39,38)(34,40,42,41)(36,43,47,46)(37,44,45,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,21)(18,20)(19,29)(22,24)(23,25)(26,30)(27,31)(28,32)(33,36,39,47)(34,37,42,45)(35,43,38,46)(40,44,41,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,15)(13,16)(17,18)(19,24)(20,25)(21,31)(22,26)(23,27)(28,30)(29,32)(33,37)(34,36)(35,44)(38,48)(39,45)(40,43)(41,46)(42,47);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,32)(18,26,27,29)(19,20,30,31)(21,24,25,28)(33,38,39,35)(34,41,42,40)(36,46,47,43)(37,48,45,44)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044773);