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 | #40948 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,10,4,8)(6,13,14,9)(11,15,12,16)(18,23)(20,27)(25,29)(28,32)(33,34,39,43)(35,40,45,37)(36,42,38,41)(44,47,46,48);(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32)(33,35,39,45)(34,40,43,37)(36,46,38,44)(41,47,42,48);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,36,39,38)(34,41,43,42)(35,44,45,46)(37,47,40,48);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,32)(28,31)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,46)(43,45);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,34)(35,37)(36,41)(38,42)(39,43)(40,45)(44,47)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,38,39,36)(34,42,43,41)(35,44,45,46)(37,47,40,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=hi,e^2=hi,f^2=i,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, 10358791);