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 | #52317 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,12)(6,9)(7,11)(8,13)(10,16)(14,15)(33,34,39,42)(35,40,45,38)(36,37,48,41)(43,47,46,44);(1,3)(2,7)(4,10)(5,9)(6,11)(8,16)(12,14)(13,15)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35,39,45)(34,40,42,38)(36,46,48,43)(37,44,41,47);(4,13)(8,12)(10,15)(14,16)(18,23)(19,25)(24,31)(26,32)(33,36,39,48)(34,41,42,37)(35,43,45,46)(38,47,40,44);(1,4,6,13)(2,8,9,12)(3,10,11,15)(5,14,7,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,36)(34,37)(35,43)(38,44)(39,48)(40,47)(41,42)(45,46);(1,5,6,7)(2,3,9,11)(4,14,13,16)(8,10,12,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37)(34,36)(35,44)(38,43)(39,41)(40,46)(42,48)(45,47);(1,3)(2,5)(4,15)(6,11)(7,9)(8,16)(10,13)(12,14)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,38)(34,35)(36,47)(37,46)(39,40)(41,43)(42,45)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=g,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, 8616372);