Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#127 |
| prime radical | #19616 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
790 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 790 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,18,15,19,13,29)(11,20,24,30,21,35,23,16)(12,17,27,33,28,34,26,40)(22,36,38,32,25,39,37,31);(9,11,13,23,15,21,14,24)(10,16,18,20,19,30,29,35)(12,25,26,38,28,22,27,37)(17,32,33,39,34,31,40,36);(1,2,3,5,4,6,7,8)(11,21)(16,30)(20,35)(22,25)(23,24)(31,32)(36,39)(37,38);(2,6)(5,8)(9,12,14,27,15,28,13,26)(10,17,18,33,19,34,29,40)(11,22,24,38,21,25,23,37)(16,31,20,36,30,32,35,39)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224816, #72224925, #72224927, … |
| maximal normal subgroup of | #72224816, #72224925, #72224927, … |
| maximal quotient of | #72287173, #72287509, #72287693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120516);