Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,27,30,31)(20,26,22,25)(23,29,32,28);(1,2)(3,4)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,31)(26,27);(10,13)(14,16)(18,24)(19,28)(20,22)(23,31)(27,32)(29,30);(2,4)(6,8)(9,11,12,15)(10,14,13,16)(19,29)(20,22)(23,31)(25,26)(27,32)(28,30)Polycyclic presentation:
2^9|a^2=i,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264432, #72264485, #72264589, … |
| maximal normal subgroup of | #72264432, #72264485, #72264589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334522);