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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1871 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)(15,19,17,20)(21,22)(23,29)(24,33)(25,35)(26,27)(28,30)(31,36)(32,34);(5,6,10,8)(7,11,9,12)(13,15,18,17)(14,19,16,20)(22,27)(23,30)(25,33)(31,34);(5,7,10,9)(6,11,8,12)(13,16,18,14)(15,19,17,20)(21,23)(22,28)(24,34)(25,32)(26,30)(27,29)(31,35)(33,36);(1,2,3,4)(5,8,10,6)(7,12,9,11)(13,17,18,15)(14,19,16,20)(21,24)(22,25)(23,31)(26,35)(27,33)(28,36)(29,32)(30,34);(5,9,10,7)(6,12,8,11)(21,25)(22,24)(23,32)(26,33)(27,35)(28,34)(29,31)(30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,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, 1101729);