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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,27)(17,23)(18,22)(19,24)(20,25)(26,28);(1,2)(3,4)(6,10)(11,12)(15,25)(16,26)(17,18)(20,21)(22,23)(27,28);(5,7,9,8)(6,11,10,12)(13,15,17,25,19,26,18,16)(14,20,22,21,24,27,23,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,26,19,25,17,15)(14,21,23,20,24,28,22,27);(5,8,9,7)(6,11,10,12)(15,26)(16,25)(20,27)(21,28)Polycyclic presentation:
2^9|c^2=fg,d^2=fgi,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 | #72257495, #72257612, #72263878, … |
| maximal normal subgroup of | #72257495, #72257612, #72263878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472704);