Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153117 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 593 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15)(7,11,18,26)(8,12,21,30)(9,13,22,14)(16,23,33,36)(17,24,34,25)(19,29,20,28)(27,32,35,31);(5,7,10,18)(6,11,15,26)(8,16,21,33)(9,17,22,34)(12,27,30,35)(13,24,14,25)(19,32,20,31)(23,28,36,29);(2,4)(5,8)(6,12)(7,16)(9,19)(10,21)(11,23)(13,29)(14,28)(15,30)(17,31)(18,33)(20,22)(24,27)(25,35)(26,36)(32,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279359, #72279943, #72279983 |
| maximal normal subgroup of | #72279359, #72279943, #72279983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60334);