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×#3021 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1928 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)(7,16)(8,11)(9,22)(10,15)(12,18)(13,20)(14,24)(17,32)(19,31)(21,30)(23,29)(25,27)(26,35)(28,36)(33,34);(1,2)(3,4)(6,11)(8,20)(12,28)(13,15)(14,30)(16,32)(17,34)(21,25)(22,27)(26,29)(31,33)(35,36);(2,4)(5,7,10,19)(6,12,15,29)(8,17,20,34)(9,23,24,18)(11,26,13,28)(14,31,22,16)(21,36,25,35)(27,33,30,32);(6,11)(8,20)(9,24)(12,26)(13,15)(14,27)(16,32)(17,34)(18,23)(22,30)(28,29)(31,33)Polycyclic presentation:
2^9|c^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 | #72337680, #72337687, #72337688, … |
| maximal normal subgroup of | #72337680, #72337687, #72337688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407053);