Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #13357 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1063 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,16)(8,13)(9,21)(10,24)(11,15)(12,18)(14,25)(17,33)(19,28)(20,32)(22,29)(23,30)(26,36)(27,34)(31,35);(1,2,4,3)(7,17)(10,25)(12,27)(14,24)(16,33)(18,34)(19,26)(20,23)(22,35)(28,36)(29,31)(30,32);(5,7)(6,12)(8,18)(9,22)(10,26)(11,19)(13,28)(14,31)(15,29)(16,21)(17,25)(20,35)(23,34)(24,27)(30,33)(32,36);(1,3,4,2)(5,8,11,9)(6,13,15,21)(7,18,19,22)(10,20,25,23)(12,28,29,16)(14,30,24,32)(17,34,26,35)(27,36,31,33)Polycyclic presentation:
2^9|b^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407982);