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 | |
3937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,9,12,14)(11,16,13,15)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(34,40)(35,42)(41,44)(43,48);(4,12)(9,14)(11,13)(15,16)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5,7,3)(2,8,6,10)(4,13,12,11)(9,15,14,16)(17,20)(18,19)(21,30)(22,23)(24,29)(25,27)(26,32)(28,31)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6,7,2)(3,10,5,8)(4,9,12,14)(11,15,13,16)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,32)(27,29)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=hi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653227);