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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,8,4)(5,13,14,9)(11,12,16,15)(20,28)(24,31)(26,32)(29,30)(33,34,39,37)(35,42,38,40)(36,44,46,41)(43,47,45,48);(1,3,6,8)(2,7,10,4)(5,11,14,16)(9,15,13,12)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,39,38)(34,40,37,42)(36,43,46,45)(41,47,44,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34,39,37)(35,40,38,42)(36,41,46,44)(43,47,45,48);(1,3)(2,7)(4,10)(5,11)(6,8)(9,15)(12,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,41)(35,43)(37,44)(38,45)(39,46)(40,47)(42,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,16,13,11)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37,39,34)(35,42,38,40)(36,41,46,44)(43,47,45,48);(1,5,6,14)(2,9,10,13)(3,11,8,16)(4,12,7,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,42,37,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044388);