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 | #55697 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,11)(10,12)(17,18)(19,25)(20,23)(21,26)(22,28)(24,29)(27,31)(30,32)(33,34,39,42)(35,38,40,36)(37,41,47,48)(43,46,45,44);(1,2,5,6)(3,7,9,14)(4,8,13,11)(10,15,12,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32)(33,35)(34,38)(36,42)(37,45)(39,40)(41,44)(43,47)(46,48);(1,2,5,6)(3,7,9,14)(4,8,13,11)(10,15,12,16)(17,20,21,29)(18,23,26,24)(19,27,28,32)(22,30,25,31)(33,36,39,38)(34,40,42,35)(37,46,47,44)(41,43,48,45);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,20)(18,24)(19,27)(21,29)(22,31)(23,26)(25,30)(28,32)(33,37,39,47)(34,41,42,48)(35,43,40,45)(36,44,38,46);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,19)(18,25)(20,27)(21,28)(22,26)(23,31)(24,30)(29,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,30)(31,32)(33,38,39,36)(34,40,42,35)(37,44,47,46)(41,43,48,45)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627861);