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 | |
2091 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)(7,11)(8,10)(9,12)(15,17)(19,20)(21,22,26,27)(23,24,30,28)(25,33,34,29)(31,36,35,32);(5,6)(7,11)(8,10)(9,12)(13,14,18,16)(15,19,17,20)(22,27)(23,30)(25,34)(32,36);(5,6,8,10)(7,9,12,11)(13,15,18,17)(14,19,16,20)(21,23,26,30)(22,28,27,24)(25,35,34,31)(29,32,33,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,17,20)(21,24,26,28)(22,23,27,30)(25,36,34,32)(29,35,33,31);(5,7)(6,9)(8,12)(10,11)(13,17,18,15)(14,19,16,20)(21,25,26,34)(22,29,27,33)(23,31,30,35)(24,32,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=g,d^2=fgh,e^2=fgh,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, 4064530);