Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130844 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8376 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61288 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,13,8,10,3,15,7,14,17,28,5,12,18,31)(4,11,19,9,22,26,32,25,21,30,23,27,20,29,16,24);(1,3)(2,9)(4,19)(5,17)(6,8)(7,18)(10,24)(11,28)(12,25)(13,30)(14,27)(15,26)(16,22)(20,32)(21,23)(29,31);(2,10,14,12)(9,24,27,25)(11,26,30,29)(13,31,28,15)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,h^b=hi,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231593, #72231708, #72231719, … |
| maximal normal subgroup of | #72231593, #72231708, #72231719, … |
| maximal quotient of | #72250429, #72250926, #72279756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38041);