Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2172 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)(5,6,10,12)(7,8,11,9)(13,14,17,20)(15,19,16,18)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,34)(35,36);(5,7,10,11)(6,8,12,9)(13,14,17,20)(15,18,16,19)(22,27)(23,29)(25,33)(31,36);(5,8,10,9)(6,7,12,11)(13,14)(15,18)(16,19)(17,20)(21,23,26,29)(22,24,27,32)(25,30,33,35)(28,31,34,36);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,16)(14,18,20,19)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);(5,9,10,8)(6,11,12,7)(13,16,17,15)(14,19,20,18)(21,25,26,33)(22,28,27,34)(23,30,29,35)(24,31,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 996625);