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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(5,13)(12,15)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(37,45)(42,46)(43,44)(47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(19,26)(22,23)(27,32)(30,31)(34,39)(40,41)(42,46)(47,48);(3,9)(4,8)(11,16)(12,15)(17,19,21,26)(18,22,25,23)(20,27,29,32)(24,30,28,31)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,35,38,36)(34,40,39,41)(37,43,45,44)(42,47,46,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=ghi,e^2=ghi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038388);