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 | #48065 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,15)(11,14)(12,16)(17,18,23,27)(19,28,29,24)(20,21,25,30)(22,31,32,26)(33,34,39,43)(35,44,41,37)(36,42,38,40)(45,47,46,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,19)(18,24)(20,22)(21,26)(23,29)(25,32)(27,28)(30,31)(33,35,39,41)(34,37,43,44)(36,45,38,46)(40,47,42,48);(1,3,6,7)(2,4,9,10)(5,11,13,14)(8,12,15,16)(17,20,23,25)(18,21,27,30)(19,22,29,32)(24,26,28,31)(33,36,39,38)(34,40,43,42)(35,45,41,46)(37,47,44,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,21)(22,26)(23,27)(25,30)(28,29)(31,32)(33,37,39,44)(34,41,43,35)(36,47,38,48)(40,46,42,45);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,15)(13,16)(17,21)(18,25)(19,26)(20,27)(22,28)(23,30)(24,32)(29,31)(33,34)(35,37)(36,40)(38,42)(39,43)(41,44)(45,47)(46,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,32)(18,26,27,31)(19,20,29,25)(21,28,30,24)(33,38,39,36)(34,42,43,40)(35,46,41,45)(37,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044774);