Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(14,18)(16,19)(21,22,25,24)(23,27,28,26);(6,9)(10,11)(13,14,17,18)(15,19,20,16)(21,23,25,28)(22,26,24,27);(7,12)(10,11)(14,18)(16,19)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,20)(14,16,18,19)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,19)(18,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4032549);