Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(13,14,19,22)(15,23,26,28)(16,24,18,21)(17,27,25,20);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,18,25,19,26,16,17)(14,20,24,23,22,27,21,28);(5,8,9,12)(6,10,7,11)(13,16,19,18)(14,21,22,24)(15,25,26,17)(20,23,27,28);(2,4)(6,7)(10,11)(14,22)(20,27)(21,24)(23,28);(13,17,16,26,19,25,18,15)(14,23,21,20,22,28,24,27)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257103, #72257120, #72257144, … |
| maximal normal subgroup of | #72257103, #72257120, #72257144, … |
| maximal quotient of | #72264873, #72308413, #72320384 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7453019);