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 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,10,4)(5,13,15,9)(11,12,16,14)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,29)(30,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,2,6,7)(3,8,10,4)(5,9,15,13)(11,14,16,12)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(3,10)(9,13)(11,16)(18,20)(24,26)(25,28)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,14)(10,16)(17,22,23,19)(18,26,20,24)(21,27,29,30)(25,31,28,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9757948);