Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128836 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,13,18)(11,21)(12,26,28,37)(14,20,15,19)(16,31)(17,30,36,29)(22,35)(23,34)(24,38)(25,39)(27,33)(32,40);(1,2,4,7,5,8,3,6)(9,11)(10,16)(12,27)(13,25)(14,24)(15,22)(17,35)(18,34)(19,33)(20,32)(21,37)(23,30)(26,39)(28,40)(29,31)(36,38);(11,22,25,24)(12,28)(16,32,34,33)(17,36)(21,35,39,38)(23,27,31,40)(26,37)(29,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,f^2=gh,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309410, #72309428, #72309432, … |
| maximal normal subgroup of | #72309410, #72309428, #72309432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36138);