Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,8)(7,12,9,11)(13,14,18,16)(15,17,20,19)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35);(5,7,10,9)(6,11,8,12)(13,15)(14,19)(16,17)(18,20)(23,30)(26,29)(31,36)(34,35);(6,8)(11,12)(14,16)(15,20)(21,22)(23,26)(24,27)(25,28)(29,30)(31,34)(32,33)(35,36);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(13,16,18,14)(15,17,20,19)(21,23,25,30)(22,26,28,29)(24,31,33,36)(27,34,32,35);(5,9,10,7)(6,11,8,12)(13,17)(14,20)(15,16)(18,19)(21,24,25,33)(22,27,28,32)(23,31,30,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=fh,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2672190);