Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119550 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
872 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 32 points:
(1,2,4,10,5,7,15,20)(3,11,14,25,13,22,27,32)(6,9,16,24,17,19,29,31)(8,21,23,26,18,28,30,12);(2,7)(3,12)(4,15)(8,22)(9,19)(11,18)(13,26)(14,28)(16,29)(21,27)(23,25)(30,32);(2,8,7,18)(3,13)(4,16)(9,11,19,22)(10,25,20,32)(12,26)(14,28)(15,29)(21,27)(23,31,30,24)Polycyclic presentation:
2^9|a^2=g,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=eh,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235157 |
| maximal normal subgroup of | #72235157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26747);