Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16)(17,18,20,23,21,24,29,32)(19,25,26,30,27,31,28,22);(2,4)(9,11)(10,14)(12,15)(13,16)(17,19,21,27)(18,22,24,30)(20,28,29,26)(23,31,32,25);(6,8)(11,15)(14,16)Polycyclic presentation:
2^9|b^2=g,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265062, #72265067, #72265104, … |
| maximal normal subgroup of | #72265062, #72265067, #72265104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371185);