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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4328 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18,24,22)(19,25,30,29)(20,31,21,26)(23,32,28,27)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,24,30)(18,25,22,29)(20,28,21,23)(26,32,31,27)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20,24,21)(18,26,22,31)(19,28,30,23)(25,32,29,27)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,21,24,20)(18,26,22,31)(19,23,30,28)(25,32,29,27)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,22,24,18)(19,29,30,25)(20,31,21,26)(23,32,28,27)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,23)(18,27)(19,20)(21,30)(22,32)(24,28)(25,26)(29,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10317043);