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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,11)(3,12,7,9)(4,6,5,10)(13,15,14,16)(17,18)(19,22)(20,25)(21,31)(23,26)(24,29)(27,32)(28,30)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,3,8,7)(2,9,11,12)(4,14,5,13)(6,15,10,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,30)(31,32)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,4,8,5)(2,10,11,6)(3,13,7,14)(9,15,12,16)(17,20,23,21)(18,25,26,31)(19,27,29,28)(22,32,24,30)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5,8,4)(2,6,11,10)(3,14,7,13)(9,16,12,15)(17,21,23,20)(18,25,26,31)(19,28,29,27)(22,32,24,30)(33,35)(34,38)(36,41)(37,42)(39,44)(40,45)(43,47)(46,48);(1,6)(2,4)(3,15)(5,11)(7,16)(8,10)(9,14)(12,13)(17,22,23,24)(18,19,26,29)(20,30,21,32)(25,28,31,27)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,7,8,3)(2,9,11,12)(4,14,5,13)(6,16,10,15)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,30)(31,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027104);