Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2340 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)(7,11)(8,10)(9,12)(13,14,18,19,17,20,15,22)(16,21,25,26,24,27,23,28);(5,7,8,12)(6,9,10,11)(13,15,17,18)(14,19,20,22)(16,23,24,25)(21,26,27,28);(2,4)(7,12)(9,11)(16,24)(21,27)(23,25)(26,28);(6,10)(9,11)(14,20)(19,22)(21,27)(26,28);(13,16,18,25,17,24,15,23)(14,21,19,26,20,27,22,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72305430, #72305438, #72305484 |
| maximal normal subgroup of | #72305430, #72305438, #72305484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284871);