Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(13,14,19,21)(15,22,16,20)(17,27,28,18)(23,24,25,26);(10,12)(13,15,19,16)(14,20,21,22)(17,23,28,25)(18,24,27,26);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,16,19,15)(14,20,21,22)(17,23,28,25)(18,26,27,24);(5,6,8,7)(13,17,19,28)(14,18,21,27)(15,23,16,25)(20,24,22,26);(1,3,4,2)(5,7,8,6)(13,18)(14,17)(15,24)(16,26)(19,27)(20,23)(21,28)(22,25)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280660, #72280783, #72280790, … |
| maximal normal subgroup of | #72280660, #72280783, #72280790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1316204);