Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133877 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1078 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,16,27,32)(11,19,22,36,20,35,26,39)(12,23,24,34,25,17,28,33)(15,29,31,37,30,38,40,21);(9,11,14,20)(10,15)(12,21,25,37)(13,26,27,22)(16,30)(17,19)(18,31)(23,35)(24,38,28,29)(32,40)(33,36)(34,39);(1,3,4,8)(2,5,6,7)(10,16)(11,20)(13,27)(17,23)(21,37)(24,28)(31,40)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=hi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,f^b=fgi,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72266411, #72268479, #72268483, … |
| maximal normal subgroup of | #72266411, #72268479, #72268483, … |
| maximal quotient of | #72266423 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41215);