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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,18,22)(15,23,16,26)(17,28,25,21)(19,27,20,24);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(15,24)(16,27)(17,25)(19,23)(20,26)(21,28);(13,15,17,24,18,16,25,27)(14,19,21,23,22,20,28,26);(2,4)(13,16,17,27,18,15,25,24)(14,20,21,26,22,19,28,23);(5,8,9,7)(6,11,12,10)(15,16)(19,20)(23,26)(24,27)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=g,d^2=g,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 | #72257108, #72257120, #72257144, … |
| maximal normal subgroup of | #72257108, #72257120, #72257144, … |
| maximal quotient of | #72264873, #72308305, #72319895 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470826);