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 | #54952 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,24)(20,29)(21,31)(22,27)(23,28)(25,30)(26,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(19,28)(20,30)(23,24)(25,29)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,6)(7,9)(8,13)(14,15)(17,19,22,28)(18,23,27,24)(20,26,30,31)(21,25,32,29)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,32)(30,31)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,25)(19,26)(21,24)(22,30)(23,32)(27,29)(28,31)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,25)(20,24)(22,32)(23,30)(27,31)(28,29)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017956);