Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455563 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2730 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16635 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,23)(3,14,18,27,19,26,17,9)(4,11,5,10,22,28,21,29)(15,30,32,25,20,24,16,31);(1,3,8,19)(2,9)(4,20,22,15)(5,16,21,32)(6,17,7,18)(10,24)(11,25)(12,14)(13,27)(23,26)(28,31)(29,30);(1,4,8,22)(2,10)(3,15,19,20)(5,7,21,6)(9,24)(11,12)(13,29)(14,25)(16,17,32,18)(23,28)(26,31)(27,30);(1,5,7,4,8,21,6,22)(2,11,12,29,13,28,23,10)(3,16,18,20,19,32,17,15)(9,25,14,30,27,31,26,24)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=g,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376135);