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 | #52394 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(33,34,39,44)(35,42,37,40)(36,41,38,43)(45,48,46,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(2,6)(7,9)(8,13)(14,15)(18,24)(23,25)(26,29)(31,32)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,37,39,35)(34,42,44,40)(36,45,38,46)(41,47,43,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,38,39,36)(34,43,44,41)(35,46,37,45)(40,48,42,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=g,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, 8062224);