Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232809 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1599 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1599 conjugacy classes of subgroups — too many to draw. Only its 202 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)(3,6)(4,7)(5,8)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,37)(23,30)(24,33)(25,35)(26,34)(28,31)(29,38)(32,39)(36,40);(2,6)(3,4)(7,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,32)(18,30)(19,31)(20,27)(24,38)(25,40)(26,39)(29,35)(33,37)(34,36);(2,7)(3,4)(6,8)(9,12)(10,17)(11,21)(13,26)(14,24)(15,25)(16,29)(18,34)(19,35)(20,36)(22,39)(23,40)(27,33)(28,38)(30,37)(31,32);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,25,26)(16,30,31,20)(17,33,35,34)(21,38,40,39)(29,36,32,37)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279810, #72280437, #72303170, … |
| maximal normal subgroup of | #72279810, #72280437, #72303170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140795);