Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| 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×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,9,7,21)(3,12,14,24,15,25,20,8)(4,16,18,10,19,28,22,26)(13,29,17,32,31,23,30,27);(1,3,7,14,6,15,5,20)(2,8,21,12,9,24,11,25)(4,17,22,31,19,30,18,13)(10,27,16,29,26,32,28,23);(2,9)(10,26)(11,21)(16,28);(1,4,6,19)(2,10,9,26)(3,13)(5,18,7,22)(8,23)(11,28,21,16)(12,27)(14,17)(15,31)(20,30)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=f,e^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,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237067, #72237071, #72237082, … |
| maximal normal subgroup of | #72237067, #72237071, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336510);