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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,10)(5,13)(7,8)(9,12)(21,29)(23,30)(26,32)(28,31)(33,34,40,38)(35,43,36,41)(37,39,48,42)(44,47,46,45);(1,2)(3,8)(4,6)(5,9)(7,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28)(33,35,40,36)(34,41,38,43)(37,44,48,46)(39,45,42,47);(1,3,6,10)(2,7,4,8)(5,11,13,16)(9,14,12,15)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31)(33,36,40,35)(34,41,38,43)(37,44,48,46)(39,47,42,45);(1,4,6,2)(3,7,10,8)(5,12,13,9)(11,14,16,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,37,40,48)(34,42,38,39)(35,44,36,46)(41,47,43,45);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,38,40,34)(35,43,36,41)(37,39,48,42)(44,45,46,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21)(18,23)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32)(33,39)(34,37)(35,45)(36,47)(38,48)(40,42)(41,44)(43,46)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034499);