Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152133 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12707 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #536 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3486 ◁ G | |
| LECS | G ▷ #536 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 641 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,12,25,39)(3,13,15,43,17,45,48,61)(4,18,21,52,20,51,55,64)(6,11,27,41,28,42,24,38)(8,29,31,47,30,57,60,54)(9,33,35,22,34,58,37,56)(14,32,46,36,19,40,53,62)(16,44,49,63,26,50,23,59);(1,3,7,17)(2,8,12,30)(4,19,20,14)(5,23,25,49)(6,26,28,16)(9,13,34,45)(10,36,39,62)(11,40,42,32)(15,27,48,24)(18,50,51,44)(21,57,55,29)(22,47,56,54)(31,38,60,41)(33,53,58,46)(35,63,37,59)(43,64,61,52);(4,20)(14,19)(21,55)(22,56)(29,57)(33,58)(46,53)(47,54)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59336);