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 | |
1926 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)(6,7)(8,12)(13,14,18,16)(15,20,17,19)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35);(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,19,17,20)(23,30)(24,33)(26,29)(27,32);(5,6,9,7)(8,10,12,11)(13,15,18,17)(14,19,16,20)(21,23)(22,26)(24,31)(25,30)(27,34)(28,29)(32,35)(33,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,18,14)(15,19,17,20)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36);(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,19,16,20)(21,24)(22,27)(23,31)(25,33)(26,34)(28,32)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=gh,d^2=gh,e^2=fgh,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, 5544660);