Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1374 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)(9,10,13,15)(11,14,12,16)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,2)(3,4)(9,11,13,12)(10,14,15,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(9,12,13,11)(10,14,15,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(2,4)(6,8)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264653, #72264706, #72264725, … |
| maximal normal subgroup of | #72264653, #72264706, #72264725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333685);