Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14,16,17)(15,19,20,18)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(6,10)(11,12)(14,17)(18,19)(21,23,24,27)(22,25,26,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19);(22,26)(25,28);(5,8,9,7)(6,12,10,11)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch128 conjugacy classes — view the full character table.
| maximal subgroup of | #72315940 |
| maximal normal subgroup of | #72315940 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515278);