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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,25,21,24,20,26)(19,27,28,23,30,32,29,31);(1,2)(3,4)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(2,4)(9,11,12,16)(10,13,14,15)(18,24)(23,31)(25,26)(27,32);(6,8)(18,25)(19,28)(20,22)(24,26)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=ef,c^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264960, #72265013, #72265080, … |
| maximal normal subgroup of | #72264960, #72265013, #72265080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331904);