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 | |
4262 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)(6,8)(7,11)(9,15)(10,16)(13,14)(18,23)(19,24)(25,30)(26,31)(33,34,39,42)(35,38,41,47)(36,37,40,46)(43,44,45,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,35)(34,38)(36,43)(37,44)(39,41)(40,45)(42,47)(46,48);(2,8)(5,7)(9,12)(14,16)(21,29)(25,30)(26,31)(27,32)(35,41)(38,47)(43,45)(44,48);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,36)(34,37)(35,43)(38,44)(39,40)(41,45)(42,46)(47,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37)(34,40)(35,44)(36,42)(38,45)(39,46)(41,48)(43,47);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,38,39,47)(34,41,42,35)(36,44,40,48)(37,45,46,43)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231533);