Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(2,3)(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,32,31,30);(1,2,4,3)(10,14)(13,15)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,3,4,2)(6,8)(11,16)(13,15)(19,27)(22,25)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=ei,b^2=f,c^2=ef,d^2=ef,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 | #72265013, #72265036, #72265143, … |
| maximal normal subgroup of | #72265013, #72265036, #72265143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339144);