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 | #45518 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4127 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)(3,11)(4,13)(5,10)(6,7)(8,14)(9,15)(12,16)(17,18,23,26)(19,22,25,32)(20,21,24,31)(27,28,29,30)(33,34)(35,36)(37,38)(39,41)(40,42)(43,44)(45,46)(47,48);(2,7)(4,15)(8,11)(10,12)(17,19)(18,22)(20,29)(21,30)(23,25)(24,27)(26,32)(28,31)(33,35)(34,37)(36,42)(38,40)(39,43)(41,45)(44,48)(46,47);(2,7)(5,16)(8,11)(9,13)(19,25)(22,32)(27,29)(28,30)(33,36,40,37)(34,35,42,38)(39,45,47,44)(41,46,48,43);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,31)(30,32)(33,37,40,36)(34,35,42,38)(39,45,47,44)(41,43,48,46);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,21)(18,24)(19,28)(20,26)(22,29)(23,31)(25,30)(27,32)(33,38,40,35)(34,37,42,36)(39,46,47,43)(41,45,48,44);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,22,23,32)(18,25,26,19)(20,28,24,30)(21,29,31,27)(33,39,40,47)(34,41,42,48)(35,43,38,46)(36,44,37,45)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306695);