Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | |
2246 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)(16,24,26,28,25,21,23,27);(5,7,8,12)(6,9,10,11)(13,15,17,18)(14,19,20,22)(16,23,25,26)(21,27,24,28);(2,4)(7,12)(9,11)(16,25)(21,24)(23,26)(27,28);(6,10)(9,11)(14,20)(19,22)(21,24)(27,28);(13,16,18,26,17,25,15,23)(14,21,19,27,20,24,22,28)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,e^a=ei,c^b=cfi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72305433, #72305484 |
| maximal normal subgroup of | #72305433, #72305484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284873);