Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2009 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)(7,10)(8,9)(11,12)(13,14,19,16)(15,22,24,28)(17,21,18,26)(20,27,25,23);(2,3)(13,15,18,23,19,24,17,27)(14,20,21,22,16,25,26,28);(1,2,4,3)(6,9)(10,12)(13,16,19,14)(15,20,24,25)(17,21,18,26)(22,27,28,23);(1,3,4,2)(13,17,19,18)(14,21,16,26)(15,23,24,27)(20,28,25,22);(5,7,8,11)(6,10,9,12)(15,24)(20,25)(22,28)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257316, #72257534, #72263840, … |
| maximal normal subgroup of | #72257316, #72257534, #72263840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374953);