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 | |
4010 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,47)(46,48);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(17,19,22,24)(18,23,26,20)(21,27,31,30)(25,32,29,28)(33,35)(34,40)(36,38)(37,46)(39,47)(41,48)(42,43)(44,45);(2,9)(3,12)(4,14)(10,16)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27)(33,36,39,37)(34,41,44,42)(35,38,47,46)(40,48,45,43);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,37,39,36)(34,42,44,41)(35,46,47,38)(40,43,45,48);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,19)(18,20)(21,30)(22,24)(23,26)(25,32)(27,31)(28,29)(33,38,39,46)(34,43,44,48)(35,36,47,37)(40,42,45,41);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)(33,34)(35,40)(36,41)(37,42)(38,48)(39,44)(43,46)(45,47)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,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, 9632028);