Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2158 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)(5,6,8,10)(7,11,12,9)(13,14)(15,16)(17,19)(18,20)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,19)(17,20)(23,30)(25,33)(27,29)(28,32);(6,10)(9,11)(14,19)(15,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(1,2)(3,4)(5,7)(6,11)(8,12)(9,10)(13,16)(14,15)(17,18)(19,20)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(5,6)(7,11)(8,10)(9,12)(13,14)(15,16)(17,19)(18,20)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5120692);