Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500019 |
| 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 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,23)(20,30,32,24)(21,26)(22,25)(27,36,31,39)(28,38)(29,37)(33,44,46,40)(34,42)(35,41)(43,47,45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,34)(22,29)(24,39)(25,37)(26,42)(27,32)(28,35)(30,36)(33,43)(38,41)(40,47)(44,48)(45,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,22,32)(18,24,25,30)(19,27,29,31)(21,33,35,46)(23,36,37,39)(26,40,41,44)(28,43,34,45)(38,47,42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(20,32)(23,37)(26,41)(27,31)(33,46)(38,42)(43,45);(2,8)(6,9)(7,16)(14,15)(17,21,22,35)(18,26,25,41)(19,28,29,34)(20,33,32,46)(23,38,37,42)(24,40,30,44)(27,43,31,45)(36,47,39,48)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436524);