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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3912 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)(3,4)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28)(33,34,39,42)(35,43,46,40)(36,47,45,37)(38,44,48,41);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,32,30,28)(33,35,39,46)(34,40,42,43)(36,38,45,48)(37,44,47,41);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,32,30,28)(33,34,39,42)(35,40,46,43)(36,47,45,37)(38,41,48,44);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,36,39,45)(34,37,42,47)(35,38,46,48)(40,44,43,41);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,21,23,19)(18,20,26,24)(22,29,31,27)(25,28,30,32)(33,37)(34,36)(35,44)(38,40)(39,47)(41,46)(42,45)(43,48);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,38)(34,41)(35,45)(36,46)(37,40)(39,48)(42,44)(43,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066192);