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 | |
3949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,6)(5,9,14,16)(11,15,12,13)(17,18)(19,20)(21,29)(22,24)(23,31)(25,27)(26,32)(28,30)(34,35)(36,43)(40,42)(44,48);(3,4)(6,8)(11,12)(13,15)(17,19,22,27)(18,20,24,25)(21,30,31,26)(23,32,29,28)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(1,3,7,4)(2,8,10,6)(5,12,14,11)(9,13,16,15)(17,18,22,24)(19,25,27,20)(21,23,31,29)(26,32,30,28)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,4,7,3)(2,6,10,8)(5,11,14,12)(9,15,16,13)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,30)(31,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47);(1,6)(2,3)(4,10)(5,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,23)(19,26)(20,28)(22,31)(24,29)(25,32)(27,30)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270926);