Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2409 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)(7,10)(8,9)(11,12)(13,14,18,22,17,21,27,24)(15,23,25,26,20,19,28,16);(6,9)(7,11);(2,4)(13,15)(14,19)(16,22)(17,20)(18,25)(21,23)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(15,20)(16,26)(19,23)(25,28);(6,9)(10,12)(13,16,18,23,17,26,27,19)(14,20,22,28,21,15,24,25)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=df,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284509, #72284588, #72284671, … |
| maximal normal subgroup of | #72284509, #72284588, #72284671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306857);