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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,26)(19,23,29,38)(20,24,33,31)(21,25,35,42)(27,36,30,39)(28,37,34,41)(32,40,46,45)(43,47,44,48);(3,7)(6,8)(10,14)(11,15)(17,19,22,29)(18,23,26,38)(20,30,33,27)(21,28,35,34)(24,39,31,36)(25,37,42,41)(32,44,46,43)(40,48,45,47);(1,2,5,4)(3,6,7,8)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,27)(21,34)(22,29)(24,36)(25,41)(26,38)(28,35)(30,33)(31,39)(32,44)(37,42)(40,48)(43,46)(45,47);(9,12)(10,15)(11,14)(13,16)(17,18)(19,23)(20,31)(21,25)(22,26)(24,33)(27,39)(28,37)(29,38)(30,36)(32,45)(34,41)(35,42)(40,46)(43,48)(44,47);(1,3)(2,6)(4,8)(5,7)(17,20)(18,24)(19,27)(21,32)(22,33)(23,36)(25,40)(26,31)(28,43)(29,30)(34,44)(35,46)(37,47)(38,39)(41,48)(42,45);(1,4,5,2)(3,8,7,6)(17,21)(18,25)(19,28)(20,32)(22,35)(23,37)(24,40)(26,42)(27,43)(29,34)(30,44)(31,45)(33,46)(36,47)(38,41)(39,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833833);