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 | |
3922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18)(19,28)(20,25)(21,31)(22,32)(23,27)(24,30)(26,29)(33,34,39,44)(35,42,37,40)(36,43,38,41)(45,47,46,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,23,30)(18,24,27,28)(20,29,31,22)(21,26,25,32)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,31)(18,25,27,21)(19,29,30,22)(24,32,28,26)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21)(18,20)(19,26)(22,28)(23,25)(24,29)(27,31)(30,32)(33,34)(35,42)(36,41)(37,40)(38,43)(39,44)(45,48)(46,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,25)(21,31)(22,26)(23,27)(28,30)(29,32)(33,37,39,35)(34,42,44,40)(36,46,38,45)(41,48,43,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,26)(19,20)(21,28)(23,29)(24,25)(27,32)(30,31)(33,38,39,36)(34,43,44,41)(35,46,37,45)(40,48,42,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=h,f^2=h,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, 8044211);