Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
2364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,8,10)(7,11,12,9)(13,14,18,17)(15,20,16,19)(22,25)(26,28);(7,12)(9,11)(13,15,18,16)(14,19,17,20)(23,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,16)(14,20,17,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,15)(14,20,17,19)(21,23,24,27)(22,26,25,28);(5,6)(7,9)(8,10)(11,12)(13,17,18,14)(15,19,16,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2725068);