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 | |
3839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(4,12)(8,10)(11,13)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(1,3,6,5)(2,8,7,10)(4,13,12,11)(9,16,14,15)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,4,6,12)(2,9,7,14)(3,11,5,13)(8,15,10,16)(17,20,22,26)(18,23,25,19)(21,28,30,32)(24,31,29,27)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,5,6,3)(2,10,7,8)(4,11,12,13)(9,15,14,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=h,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, 8066269);