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 | |
3831 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(34,40)(35,45)(37,43)(42,47);(2,6)(3,5)(4,12)(15,16)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8937324);