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 | |
4154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,38,41)(35,40,44,39)(36,45,46,37)(42,43,48,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,35,38,44)(34,39,41,40)(36,42,46,48)(37,43,45,47);(3,11)(7,8)(12,16)(13,14)(20,28)(23,31)(27,32)(29,30)(33,35,38,44)(34,39,41,40)(36,42,46,48)(37,43,45,47);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,22,28)(18,23,26,31)(19,27,25,32)(21,30,24,29)(33,35)(34,40)(36,42)(37,47)(38,44)(39,41)(43,45)(46,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,36,38,46)(34,37,41,45)(35,42,44,48)(39,43,40,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,37)(34,36)(35,43)(38,45)(39,42)(40,48)(41,46)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641158);