Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(14,19)(15,22)(16,18)(20,28)(23,27)(24,25);(1,2)(3,4)(6,9)(10,11)(13,14,17,19)(15,23,24,28)(16,21,18,26)(20,25,27,22);(13,15,16,25,17,24,18,22)(14,20,26,28,19,27,21,23);(2,4)(13,15,16,25,17,24,18,22)(14,20,26,28,19,27,21,23);(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,21,19,26)(15,22,24,25)(20,28,27,23)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257120, #72257122, #72257188, … |
| maximal normal subgroup of | #72257120, #72257122, #72257188, … |
| maximal quotient of | #72264858, #72308323, #72320024 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470162);