Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458069 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2653 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16416 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,14,23)(4,16,17,8)(5,11)(6,10)(9,18,27,19)(12,25)(13,21)(15,24,30,31)(20,26)(22,28)(29,32);(1,3)(2,7,10,23)(4,13)(5,15)(6,14)(8,25,16,21)(9,22,27,32)(11,24,26,31)(12,17)(18,29)(19,28)(20,30);(1,4)(2,8)(3,12,14,13)(5,19)(6,17)(7,21,23,25)(9,26)(10,16)(11,27)(15,29,30,28)(18,20)(22,31,32,24);(1,4)(2,9)(3,13)(5,19)(6,17)(7,22)(8,11)(10,27)(12,14)(15,28)(16,26)(18,20)(21,31)(23,32)(24,25)(29,30)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369276);