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 | #52313 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4319 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)(17,18,23,25)(19,20,21,24)(22,30,31,26)(27,32,29,28)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,6)(7,9)(8,13)(14,15)(18,25)(20,24)(26,30)(28,32)(33,35,38,36)(34,40,42,39)(37,43,46,44)(41,48,45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,19)(18,20,25,24)(22,29,31,27)(26,28,30,32)(33,36,38,35)(34,39,42,40)(37,44,46,43)(41,47,45,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062216);