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 | #40796 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4095 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)(3,9)(4,7)(5,10)(6,12)(8,13)(11,15)(14,16)(18,23)(19,28)(20,27)(30,31)(33,34,39,43)(35,40,45,47)(36,46,38,41)(37,42,48,44);(1,3)(2,6)(4,11)(5,12)(7,14)(8,15)(9,10)(13,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,31)(30,32)(33,35,39,45)(34,40,43,47)(36,42,38,44)(37,46,48,41);(1,4,5,13)(2,7,10,8)(3,11,12,16)(6,14,9,15)(17,19,22,28)(18,24,23,25)(20,29,27,32)(21,30,26,31)(33,36,39,38)(34,41,43,46)(35,44,45,42)(37,40,48,47);(1,4)(2,8)(3,11)(5,13)(6,15)(7,10)(9,14)(12,16)(17,19)(18,25)(20,29)(21,31)(22,28)(23,24)(26,30)(27,32)(33,37,39,48)(34,42,43,44)(35,41,45,46)(36,47,38,40);(1,3)(2,9)(4,11)(5,12)(6,10)(7,15)(8,14)(13,16)(17,20)(18,26)(19,29)(21,23)(22,27)(24,31)(25,30)(28,32)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,48)(45,47);(1,2)(3,6)(4,7)(5,10)(8,13)(9,12)(11,14)(15,16)(17,21,22,26)(18,27,23,20)(19,30,28,31)(24,32,25,29)(33,38,39,36)(34,41,43,46)(35,44,45,42)(37,47,48,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,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, 8627798);