Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153472 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13561 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | #18547 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3152 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,8)(4,15,18,7)(5,11)(6,10,19,26)(9,16,22,20)(12,27,28,23)(13,25)(14,24,29,32)(17,31,30,21);(2,7)(4,16)(8,23)(9,11)(10,15)(12,17)(18,20)(21,25)(22,26)(24,27)(28,30)(31,32);(1,3,5,13)(2,8,11,25)(4,17,16,12)(6,14,19,29)(7,21,9,23)(10,24,26,32)(15,31,22,27)(18,30,20,28)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=chi,d^b=di,e^b=efh,f^b=fhi,g^b=ghi,d^c=di,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72306810, #72306818, #72306840, … |
| maximal normal subgroup of | #72306810, #72306818, #72306840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60669);