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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1804 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)(13,14,17,20)(15,21,22,19)(16,18,25,26)(23,28,27,24);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,22)(14,19,20,21)(16,24,25,28)(18,27,26,23);(13,16,18,20,17,25,26,14)(15,21,23,24,22,19,27,28);(2,4)(13,16,18,20,17,25,26,14)(15,21,23,24,22,19,27,28);(5,8,9,7)(6,11,12,10)(14,20)(16,25)(19,21)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=fi,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 | #72308746 |
| maximal normal subgroup of | #72308746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470823);