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 | |
4121 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34,38,43)(35,41,44,39)(36,40,45,46)(37,42,47,48);(2,7)(3,11)(9,12)(10,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,40)(38,44)(41,43)(45,48)(46,47);(2,7)(3,11)(9,12)(10,16)(19,26)(22,23)(27,32)(30,31)(33,36,38,45)(34,40,43,46)(35,42,44,48)(37,41,47,39);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,39)(38,45)(41,47)(43,46)(44,48);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,35)(34,41)(36,42)(37,46)(38,44)(39,43)(40,47)(45,48);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,37,38,47)(34,42,43,48)(35,40,44,46)(36,41,45,39)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=h,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, 8635712);