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 | |
4108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,32,30)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(2,7)(3,10)(9,12)(11,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,38,44)(34,39,42,43)(36,41,45,46)(37,40,47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,22,29)(18,23,26,24)(20,27,25,32)(21,28,31,30)(33,36,38,45)(34,40,42,48)(35,41,44,46)(37,39,47,43);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,24)(20,27)(21,30)(22,29)(23,26)(25,32)(28,31)(33,36)(34,40)(35,41)(37,43)(38,45)(39,47)(42,48)(44,46);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20)(18,21)(19,27)(22,25)(23,28)(24,30)(26,31)(29,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,32)(24,27)(29,30)(33,37,38,47)(34,41,42,46)(35,40,44,48)(36,39,45,43)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,e^2=h,f^2=g,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, 8649141);