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 | |
4041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,13,14,8)(4,5,11,10)(6,16,15,12)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34,39,37)(35,42,44,40)(36,45,38,41)(43,47,46,48);(1,3)(2,8)(4,15)(5,16)(6,11)(7,14)(9,13)(10,12)(17,19,23,31)(18,24,28,29)(20,30,21,22)(25,27,26,32)(33,35)(34,40)(36,43)(37,42)(38,46)(39,44)(41,47)(45,48);(2,9)(3,14)(4,11)(12,16)(17,20,23,21)(18,25,28,26)(19,30,31,22)(24,27,29,32)(33,36,39,38)(34,41,37,45)(35,43,44,46)(40,47,42,48);(1,4,7,11)(2,10,9,5)(3,15,14,6)(8,12,13,16)(17,21,23,20)(18,26,28,25)(19,22,31,30)(24,32,29,27)(33,37,39,34)(35,42,44,40)(36,41,38,45)(43,47,46,48);(1,5,7,10)(2,11,9,4)(3,12,14,16)(6,8,15,13)(17,22)(18,27)(19,20)(21,31)(23,30)(24,26)(25,29)(28,32)(33,38,39,36)(34,41,37,45)(35,43,44,46)(40,48,42,47);(1,6,7,15)(2,12,9,16)(3,11,14,4)(5,8,10,13)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,35)(34,40)(36,43)(37,42)(38,46)(39,44)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9630360);