Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1924 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)(7,12)(8,11)(9,10)(13,14,16,19)(15,18,20,17)(22,26)(23,29)(28,33)(31,35);(6,9)(11,12)(13,15,16,20)(14,17,19,18)(24,32)(28,33)(31,35)(34,36);(5,7,10,8)(6,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,14)(15,17)(16,19)(18,20)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,9,10,6)(7,12,8,11)(13,15)(14,18)(16,20)(17,19)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5545075);