Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418498 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15399 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,14,12)(3,13,17,30,19,29,18,8)(4,5,21,25,22,9,23,26)(15,31,32,16,28,24,20,27);(1,3,14,17,7,19,6,18)(2,8,12,13,11,30,10,29)(4,20,23,15,22,32,21,28)(5,24,26,27,9,31,25,16);(1,4)(2,9)(3,15)(5,11)(6,21)(7,22)(8,27)(10,26)(12,25)(13,31)(14,23)(16,30)(17,32)(18,20)(19,28)(24,29);(1,5,7,9)(2,4,11,22)(3,16)(6,25,14,26)(8,28)(10,21,12,23)(13,20)(15,30)(17,24)(18,31)(19,27)(29,32)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319831, #72319908, #72320000, … |
| maximal normal subgroup of | #72319831, #72319908, #72320000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336512);