Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2477 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15670 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,22,24,15,8,16,25)(4,17,19,28,20,9,21,27)(14,29,31,32,30,23,18,26);(1,3,5,16,7,15,6,22)(2,8,12,24,11,13,10,25)(4,18,19,30,20,31,21,14)(9,26,27,23,17,32,28,29);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,23)(10,27)(11,17)(12,28)(13,29)(15,30)(16,31)(18,22)(24,26)(25,32);(2,10,11,12)(8,24,13,25)(9,27,17,28)(23,26,29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338513, #72338519, #72338527, … |
| maximal normal subgroup of | #72338513, #72338519, #72338527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348099);