Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,12)(13,14,17,19)(15,16,20,18)(23,27)(26,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15,17,20)(14,18,19,16)(22,25)(26,28);(1,2,3,4)(7,11)(10,12)(14,19)(16,18)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,18)(14,15,19,20)(21,23)(22,26)(24,27)(25,28);(5,6)(7,12)(8,9)(10,11)(13,15)(14,16)(17,20)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ghi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3014027);