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 | #55684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4077 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)(3,8)(4,5)(6,7)(9,13)(10,16)(11,14)(12,15)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,7)(4,13)(8,15)(10,14)(17,19,22,24)(18,23,26,20)(21,27,29,31)(25,30,32,28)(33,35,39,44)(34,40,43,48)(36,47,41,38)(37,42,46,45);(2,7)(5,9)(8,15)(11,16)(21,29)(25,32)(27,31)(28,30)(35,44)(38,47)(40,48)(42,45);(1,3,6,12)(2,8,7,15)(4,14,13,10)(5,11,9,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,36)(34,37)(35,38)(39,41)(40,45)(42,48)(43,46)(44,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,23)(25,31)(27,32)(29,30)(33,37)(34,41)(35,45)(36,43)(38,48)(39,46)(40,47)(42,44);(1,5,6,9)(2,4,7,13)(3,11,12,16)(8,14,15,10)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,38)(34,42)(35,36)(37,48)(39,47)(40,46)(41,44)(43,45)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8314065);