Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245168 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36703 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1685 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1685 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,16)(5,8)(6,11)(7,14)(9,17)(10,20)(13,27)(15,25)(18,23)(19,26)(21,32)(22,29)(24,31)(28,30);(4,17)(7,21)(9,16)(12,27)(14,18)(19,31)(22,23)(24,26)(25,28)(29,32);(1,3,6,15)(2,7,11,23)(4,18,17,14)(5,13,20,30)(8,22,10,21)(9,25,16,12)(19,32,31,29)(24,28,26,27);(2,8)(4,19)(7,22)(10,11)(14,29)(17,31)(18,32)(21,23)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72341223, #72341226, #72341235 |
| maximal normal subgroup of | #72341223, #72341226, #72341235 |
| maximal quotient of | #72227855 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152376);