Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,18,21,25)(16,19,24,28)(22,26,23,27);(2,3)(6,9)(10,11)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,24)(27,28);(1,2,4,3)(15,21)(18,25)(22,23)(26,27);(1,3,4,2)(15,22)(16,24)(18,26)(19,28)(21,23)(25,27);(5,7,8,12)(6,10,9,11)(15,21)(18,25)(22,23)(26,27)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282625 |
| maximal normal subgroup of | #72282625 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469952);