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 | #47992 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,14)(5,9)(6,10)(7,13)(11,12)(15,16)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(34,39)(35,41)(37,44)(42,45);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,34,38,39)(35,40,41,48)(36,44,46,37)(42,43,45,47);(4,9)(5,14)(11,15)(12,16)(19,27)(20,29)(23,30)(24,32)(33,35,38,41)(34,40,39,48)(36,42,46,45)(37,47,44,43);(1,4,6,9)(2,5,10,14)(3,11,13,15)(7,12,8,16)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,35)(34,40)(36,45)(37,43)(38,41)(39,48)(42,46)(44,47);(1,3)(2,8)(4,15)(5,12)(6,13)(7,10)(9,11)(14,16)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,36)(34,37)(35,42)(38,46)(39,44)(40,47)(41,45)(43,48);(1,5)(2,9)(3,12)(4,10)(6,14)(7,15)(8,11)(13,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,37)(34,36)(35,43)(38,44)(39,46)(40,45)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8812428);