Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62781 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,22,26)(19,23,29,38)(20,24,32,41)(21,33,35,25)(27,39,30,36)(28,42,34,37)(31,45,46,40)(43,47,44,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,23)(20,30)(21,34)(22,29)(24,39)(25,42)(26,38)(27,32)(28,35)(31,43)(33,37)(36,41)(40,47)(44,46)(45,48);(3,9)(6,13)(10,15)(14,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,36)(25,40)(26,41)(28,43)(29,30)(33,45)(34,44)(35,46)(37,47)(38,39)(42,48);(3,9)(6,13)(10,15)(14,16)(17,21,22,35)(18,25,26,33)(19,28,29,34)(20,31,32,46)(23,37,38,42)(24,40,41,45)(27,43,30,44)(36,47,39,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,26)(23,38)(24,41)(25,33)(36,39)(37,42)(40,45)(47,48)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403468);