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 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1865 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1865 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)(7,19)(8,24)(9,17)(10,14)(11,28)(12,18)(13,22)(15,26)(16,25)(20,32)(21,31)(23,30)(27,33)(29,34)(35,36);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,27)(11,29)(12,23)(14,19)(15,31)(16,30)(17,32)(18,33)(22,25)(24,34)(26,35)(28,36);(2,4)(5,8,12,26)(6,14,18,16)(7,20,23,35)(9,10,11,25)(13,30,33,19)(15,28,24,17)(21,27,29,22)(31,32,34,36);(5,9,12,11)(6,15,18,24)(7,21,23,29)(8,25,26,10)(13,31,33,34)(14,17,16,28)(19,32,30,36)(20,22,35,27)Polycyclic presentation:
2^9|c^2=i,d^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,d^c=di,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 | #72212257, #72212272, #72213178, … |
| maximal normal subgroup of | #72212257, #72212272, #72213178, … |
| maximal quotient of | #72212208, #72213520, #72266753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407061);