Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,15)(16,18,20,19)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35);(6,8)(10,11)(13,15,17,14)(16,18,20,19)(23,30)(24,33)(26,29)(27,32);(6,8)(7,12)(13,16,17,20)(14,18,15,19)(21,23)(22,26)(24,31)(25,30)(27,34)(28,29)(32,35)(33,36);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36);(5,8,9,6)(7,10,12,11)(21,24)(22,27)(23,31)(25,33)(26,34)(28,32)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4044311);