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 | |
4088 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)(3,10)(4,9)(5,7)(6,8)(11,16)(12,14)(13,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,42)(36,38)(37,41)(39,43)(40,47)(44,48)(45,46);(1,3,6,5)(2,7,8,10)(4,12,13,11)(9,16,15,14)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,8)(7,10)(9,15)(14,16)(18,20)(19,26)(25,28)(27,29)(33,36,39,37)(34,38,43,41)(35,44,47,45)(40,46,42,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,16)(13,15)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,4,6,13)(2,9,8,15)(3,11,5,12)(7,14,10,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,35)(34,42)(36,44)(37,45)(38,48)(39,47)(40,43)(41,46);(1,5,6,3)(2,10,8,7)(4,11,13,12)(9,14,15,16)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=g,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, 9633813);