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 | #55014 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,25)(19,20,29,24)(21,26,31,30)(22,32,28,27)(33,34,38,42)(35,39,45,40)(36,41,47,37)(43,48,46,44);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,19,23,29)(18,24,25,20)(21,28,31,22)(26,32,30,27)(33,34)(35,39)(36,37)(38,42)(40,45)(41,47)(43,44)(46,48);(2,6)(3,10)(4,13)(15,16)(18,25)(20,24)(21,31)(22,28)(33,35,38,45)(34,39,42,40)(36,46,47,43)(37,48,41,44);(1,3)(2,7)(4,14)(5,10)(6,9)(8,16)(11,13)(12,15)(17,20,23,24)(18,19,25,29)(21,32,31,27)(22,30,28,26)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,48)(43,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,21)(18,26)(19,28)(20,27)(22,29)(23,31)(24,32)(25,30)(33,36)(34,41)(35,43)(37,42)(38,47)(39,48)(40,44)(45,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22,23,28)(18,27,25,32)(19,21,29,31)(20,30,24,26)(33,37,38,41)(34,36,42,47)(35,44,45,48)(39,43,40,46)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8965818);