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 | |
4091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,10,4)(6,13,15,9)(11,12,14,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34)(35,44)(36,45)(37,38)(39,46)(40,43)(41,48)(42,47);(1,3)(2,8)(4,5)(6,14)(7,10)(9,12)(11,15)(13,16)(33,35,40,38)(34,37,43,44)(36,46,48,42)(39,45,47,41);(2,5)(3,10)(9,13)(11,14)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36)(34,41)(35,42)(37,47)(38,46)(39,44)(40,48)(43,45);(1,4)(2,3)(5,10)(6,12)(7,8)(9,11)(13,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,37,40,44)(34,35,43,38)(36,47,48,39)(41,42,45,46);(1,5,7,2)(3,4,10,8)(6,13,15,9)(11,12,14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,38,40,35)(34,37,43,44)(36,46,48,42)(39,41,47,45);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,16)(10,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,47)(34,42,43,46)(35,45,38,41)(36,44,48,37)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=h,f^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, 10247470);