Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 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)(17,18,24,28)(19,31,33,40)(20,34,38,46)(21,27,25,41)(22,30,23,29)(26,42,44,37)(32,36,35,39)(43,47,45,48);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,35)(21,28)(22,41)(23,27)(24,33)(26,39)(29,40)(30,31)(32,38)(34,47)(36,44)(37,45)(42,43)(46,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,26)(19,32)(21,39)(22,37)(23,42)(24,38)(25,36)(27,45)(28,44)(29,34)(30,46)(31,47)(33,35)(40,48)(41,43);(3,9)(6,13)(10,15)(14,16)(17,21)(18,27)(19,29)(20,36)(22,31)(23,40)(24,25)(26,43)(28,41)(30,33)(32,46)(34,35)(37,48)(38,39)(42,47)(44,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,28)(19,33)(26,44)(27,41)(29,30)(32,35)(34,46)(43,45)Polycyclic presentation:
2^9|a^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7073035);