Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
2236 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)(7,8)(9,11)(10,12)(14,18)(15,16)(21,22,24,26)(23,27,28,25);(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,20,16,19)(21,22,24,26)(23,27,28,25);(6,11)(8,10)(13,15,17,16)(14,19,18,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,17,15)(14,20,18,19);(5,8)(6,12)(7,11)(9,10)(13,14)(15,20)(16,19)(17,18)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fgh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3950728);