Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111253 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #34643 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
912 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,4,7,5,3,8,6)(9,10,14,20,15,18,13,19)(11,21,23,35,24,16,29,31)(12,25,28,17,26,34,27,33)(22,36,38,30,37,40,39,32);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,32)(18,21)(19,31)(20,35)(25,36)(26,37)(27,38)(28,39)(30,33)(34,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,27)(15,26)(16,30)(18,33)(19,34)(20,25)(21,32)(23,39)(24,37)(29,38)(31,36)(35,40)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dh,e^c=eh,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302659, #72302768 |
| maximal normal subgroup of | #72302659, #72302768 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18452);