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 |
| prime radical | #39277 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,10,9)(4,7,13,12)(11,14,16,15)(17,18)(19,28)(20,30)(21,31)(22,27)(23,25)(24,29)(26,32)(33,34)(35,40)(36,37)(38,43)(39,41)(42,47)(44,48)(45,46);(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,13)(14,16)(17,19,23,29)(18,24,25,28)(20,31,32,22)(21,30,27,26)(33,35,39,44)(34,40,41,48)(36,45,42,43)(37,46,47,38);(3,10)(6,9)(11,16)(14,15)(18,25)(21,27)(24,28)(26,30)(34,41)(35,44)(37,47)(43,45);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,15)(14,16)(17,20,23,32)(18,26,25,30)(19,22,29,31)(21,24,27,28)(33,36,39,42)(34,37,41,47)(35,43,44,45)(38,48,46,40);(1,3,5,10)(2,6,8,9)(4,11,13,16)(7,14,12,15)(17,21,23,27)(18,22,25,31)(19,26,29,30)(20,24,32,28)(33,37)(34,42)(35,38)(36,41)(39,47)(40,45)(43,48)(44,46);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,22)(18,27)(19,20)(21,25)(23,31)(24,26)(28,30)(29,32)(33,38)(34,43)(35,37)(36,40)(39,46)(41,45)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495067);