Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1924 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1924 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(5,6)(7,17)(8,13)(9,12)(10,16)(11,18)(14,22)(15,24)(19,32)(20,31)(21,28)(23,29)(25,30)(26,34)(27,36)(33,35);(1,2)(3,4)(5,7,10,20)(6,11,16,28)(8,21,22,18)(9,19,24,35)(12,27,15,26)(13,17,14,31)(23,36,25,34)(29,33,30,32);(2,4)(6,12)(9,24)(11,26)(13,29)(14,30)(15,16)(17,32)(19,35)(23,25)(27,28)(31,33)(34,36);(6,13)(8,22)(9,24)(11,17)(12,29)(14,16)(15,30)(18,21)(19,35)(26,32)(27,33)(28,31)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72210404, #72210417, #72213178, … |
| maximal normal subgroup of | #72210404, #72210417, #72213178, … |
| maximal quotient of | #72210362, #72213149, C2×#499856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407052);