Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1270 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 148 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)(7,11)(8,12)(9,14)(10,13)(15,33)(16,25)(17,26)(18,27)(19,32)(20,28)(21,31)(22,29)(23,30)(24,34)(35,36);(1,2,4,3)(5,7,10,16)(6,11,13,25)(8,18,20,35)(9,21,23,17)(12,27,28,36)(14,31,30,26)(15,22,34,19)(24,32,33,29);(5,8)(6,12)(7,15,16,34)(9,22,23,19)(10,20)(11,24,25,33)(13,28)(14,32,30,29)(17,35)(18,21)(26,36)(27,31);(1,3,4,2)(6,13)(11,25)(12,28)(14,30)(24,33)(26,31)(27,36)(29,32)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410396);