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 | #48185 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,8)(5,9,13,12)(11,14,16,15)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,29,31,27)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,15)(13,16)(17,19,23,30)(18,24,28,32)(20,29,26,27)(21,31,25,22)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(3,10)(7,8)(11,16)(14,15)(17,20,23,26)(18,25,28,21)(19,27,30,29)(22,24,31,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20)(18,25)(19,29)(21,28)(22,24)(23,26)(27,30)(31,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4,6,2)(3,8,10,7)(5,9,13,12)(11,14,16,15)(17,21,23,25)(18,26,28,20)(19,22,30,31)(24,29,32,27)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,27)(19,21)(20,24)(23,31)(25,30)(26,32)(28,29)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9111030);