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 | #57270 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,14,16,15)(20,28)(23,30)(26,32)(29,31)(33,34)(35,44)(36,40)(37,45)(38,43)(39,46)(41,47)(42,48);(2,6)(7,9)(8,12)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,39)(36,45)(37,41)(38,46)(40,48)(42,47)(43,44);(2,6)(7,9)(8,12)(14,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,36,38,47)(34,40,43,41)(35,45,46,42)(37,39,48,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36)(34,41)(35,45)(37,39)(38,47)(40,43)(42,46)(44,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,34)(35,39)(36,40)(37,42)(38,43)(41,47)(44,46)(45,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37,38,48)(34,42,43,45)(35,41,46,40)(36,39,47,44)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,e^2=h,f^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, 8064357);