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 | #44078 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,32)(29,31)(33,34,39,43)(35,44,46,40)(36,41,47,38)(37,48,45,42);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,31)(27,30)(28,32)(33,35)(34,40)(36,45)(37,47)(38,42)(39,46)(41,48)(43,44);(2,6)(3,10)(4,13)(15,16)(18,22)(20,30)(21,26)(27,29)(33,36,39,47)(34,41,43,38)(35,45,46,37)(40,48,44,42);(1,3)(2,7)(4,14)(5,10)(6,9)(8,16)(11,13)(12,15)(17,20)(18,24)(19,27)(21,32)(22,28)(23,30)(25,29)(26,31)(33,37,39,45)(34,42,43,48)(35,36,46,47)(38,40,41,44);(1,4)(2,8)(3,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,21,23,26)(18,25,22,19)(20,31,30,32)(24,27,28,29)(33,36)(34,41)(35,37)(38,43)(39,47)(40,42)(44,48)(45,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22,23,18)(19,26,25,21)(20,28,30,24)(27,31,29,32)(33,38,39,41)(34,36,43,47)(35,42,46,48)(37,40,45,44)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966546);