Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,10)(7,11,8,12)(13,14,19,17)(15,22,25,21)(16,18,28,20)(23,26,27,24);(2,3)(6,10)(11,12)(13,15,19,25)(14,21,17,22)(16,27,28,23)(18,26,20,24);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,19,28)(14,20,17,18)(15,23,25,27)(21,26,22,24);(1,3,4,2)(13,17,19,14)(15,23,25,27)(16,18,28,20)(21,24,22,26);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,16,17,28)(15,24,25,26)(21,23,22,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72257226, #72257580, #72263840, … |
| maximal normal subgroup of | #72257226, #72257580, #72263840, … |
| maximal quotient of | #72264168 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471986);