Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629016 |
| 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×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(15,22)(16,24)(17,26)(20,25)(21,27);(5,7,8,11)(6,9,10,12)(13,14,17,19,18,21,26,27)(15,20,23,16,22,24,28,25);(6,10)(9,12)(13,15)(14,20)(16,19)(17,23)(18,22)(21,24)(25,27)(26,28);(2,4)(13,16,17,24,18,25,26,20)(14,15,19,23,21,22,27,28);(14,21)(15,22)(19,27)(23,28)Polycyclic presentation:
2^9|b^2=fgi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250081, #72250109, #72250172, … |
| maximal normal subgroup of | #72250081, #72250109, #72250172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536952);