Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #23872 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,8)(4,9,15,7)(5,10)(6,11)(12,24)(13,23,16,20)(14,25)(17,26,31,21)(18,28)(19,27)(22,29,32,30);(2,7)(3,12)(4,15)(5,18)(8,22)(9,11)(10,26)(13,30)(14,19)(16,29)(20,24)(21,28)(23,27)(25,32);(1,3,6,14)(2,8,11,25)(4,16,15,13)(5,19,18,12)(7,20,9,23)(10,27,28,24)(17,29,31,30)(21,32,26,22);(2,7)(4,15)(8,20)(9,11)(10,21)(13,16)(17,31)(22,24)(23,25)(26,28)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237519, #72237525, #72237551, … |
| maximal normal subgroup of | #72237519, #72237525, #72237551, … |
| maximal quotient of | #72311967, #72312018, #72312034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408306);