Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153629 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13673 |
| commutator, Frattini, cosocle | #492 | C23 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,8)(5,11)(6,21)(7,12)(10,17)(13,26)(14,22)(15,25)(16,30)(18,31)(19,23)(20,27)(24,28)(29,32);(2,8)(3,13)(4,17)(9,24)(10,12)(11,27)(15,16)(19,20)(21,23)(22,26)(25,31)(30,32);(1,3,7,13)(2,9,12,26)(4,18,17,29)(5,16,6,15)(8,22,10,24)(11,25,21,30)(14,20,28,19)(23,31,27,32)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=efgh,g^b=gh,d^c=df,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72231248, C2×#153630, #72283015, … |
| maximal normal subgroup of | #72231248, C2×#153630, #72283015, … |
| maximal quotient of | C2×#153630, #72283763, #72309219 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60826);