Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3606 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)(6,8)(9,10)(11,12)(13,14,17,20)(15,18,22,26)(16,23,24,19)(21,27,28,25);(2,4)(5,6,7,8)(13,15,17,22)(14,18,20,26)(16,21,24,28)(19,25,23,27);(9,10,11,12)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,27)(24,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28);(13,16,17,24)(14,19,20,23)(15,21,22,28)(18,25,26,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280694 |
| maximal normal subgroup of | #72280694 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837448);