Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251935 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1542 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1542 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,23)(17,29)(20,27)(21,37)(22,33)(24,34)(25,32)(28,31)(30,39)(35,38)(36,40);(1,3,4,7)(2,5,6,8)(9,10)(11,20,22,15)(12,16)(13,17)(14,18)(19,23,33,27)(21,36,38,30)(24,31)(25,32)(26,29)(28,34)(35,40,37,39);(5,8)(9,11)(10,15)(12,21)(13,27)(14,22)(16,30)(17,33)(18,20)(19,26)(23,29)(24,40)(25,38)(28,39)(31,35)(32,36)(34,37);(9,12,14,25)(10,16,18,32)(11,21,22,38)(13,28,29,24)(15,30,20,36)(17,34,26,31)(19,35,33,37)(23,40,27,39)Polycyclic presentation:
2^9|b^2=ei,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72303845 |
| maximal normal subgroup of | #72303845 |
| maximal quotient of | #72304003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159148);