Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,20,23,28,31)(19,25,27,32,26,22,29,30);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16);(10,14)(11,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,23)(19,26)(24,31)(27,29);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72283545, #72283584, #72283620, … |
| maximal normal subgroup of | #72283545, #72283584, #72283620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315392);