Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3452 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)(6,9)(10,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(2,4)(15,21)(18,24)(22,27)(25,28);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,19)(15,21)(20,26)(22,27)Polycyclic presentation:
2^9|a^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=df,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72305354, #72305355, #72305375, … |
| maximal normal subgroup of | #72305354, #72305355, #72305375, … |
| maximal quotient of | #72305666, #72305746, #72305921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286691);