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 | |
4110 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(33,34,39,42)(35,40,37,45)(36,38,41,48)(43,44,46,47);(2,8)(5,11)(9,15)(13,16)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(35,37)(40,45)(43,46)(44,47);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,35,39,37)(34,40,42,45)(36,43,41,46)(38,44,48,47);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,36)(34,38)(35,43)(37,46)(39,41)(40,44)(42,48)(45,47);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,37,39,35)(34,40,42,45)(36,43,41,46)(38,47,48,44);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,38)(34,41)(35,44)(36,42)(37,47)(39,48)(40,46)(43,45)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=i,f^2=h,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, 10347231);