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 | |
3957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,22,25)(19,23,20,27)(21,24,30,29)(26,31,28,32)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,19,22,20)(18,23,25,27)(21,26,30,28)(24,31,29,32)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,30)(26,32)(28,31)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,20,22,19)(18,23,25,27)(21,28,30,26)(24,31,29,32)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,22,30)(18,24,25,29)(19,26,20,28)(23,31,27,32)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,e^2=g,f^2=g,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, 10315558);