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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2192 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)(5,6,9,11)(7,12,8,10)(13,14,18,17)(15,20,16,19)(21,22,26,27)(23,29,30,24)(25,33,34,28)(31,32,35,36);(5,7,9,8)(6,10,11,12)(13,15,18,16)(14,19,17,20)(22,27)(23,30)(25,34)(32,36);(7,8)(10,12)(13,15,18,16)(14,20,17,19)(21,23,26,30)(22,24,27,29)(25,35,34,31)(28,36,33,32);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16,18,15)(14,19,17,20)(21,24)(22,23)(25,36)(26,29)(27,30)(28,35)(31,33)(32,34);(5,8,9,7)(6,10,11,12)(13,17,18,14)(15,20,16,19)(21,25,26,34)(22,28,27,33)(23,31,30,35)(24,32,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgh,d^2=fh,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, 4049432);