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 | |
4038 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)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(34,39)(35,42)(36,44)(45,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(19,23)(20,26)(27,29)(28,31)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(3,12)(6,15)(8,16)(10,13)(17,19,22,23)(18,20,25,26)(21,29,30,27)(24,31,32,28)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,32)(26,31)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,e^2=g,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, 8313878);