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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1954 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)(5,6,9,7)(8,11,12,10)(13,14,17,19)(15,16,20,18)(21,22,26,27)(23,31,34,28)(24,29,30,25)(32,33,35,36);(8,12)(10,11)(13,14,17,19)(15,16,20,18)(22,27)(25,29)(28,31)(33,36);(6,7)(10,11)(13,15,17,20)(14,18,19,16)(21,23)(22,28)(24,35)(25,36)(26,34)(27,31)(29,33)(30,32);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,24)(22,29)(23,32)(25,27)(26,30)(28,36)(31,33)(34,35);(5,8,9,12)(6,10,7,11)(13,16)(14,15)(17,18)(19,20)(21,25,26,29)(22,30,27,24)(23,33,34,36)(28,35,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=fgh,e^2=gh,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, 4068449);