Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23912 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1225 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1225 conjugacy classes of subgroups — too many to draw. Only its 140 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)(7,12)(8,19)(9,14,23,11)(10,15)(13,22,32,25)(16,33)(17,28,24,26)(18,29)(20,30)(21,36)(27,34,31,35);(1,2)(3,4)(6,11)(8,20)(9,23)(12,26)(13,30)(14,15)(16,21)(17,24)(19,32)(27,36)(28,29)(31,33);(2,4)(5,7,10,18)(6,12,15,29)(8,21,20,16)(9,24,23,17)(11,26,14,28)(13,31,32,27)(19,36,30,33)(22,34,25,35);(5,8,10,20)(6,13,15,32)(7,16,18,21)(9,25,23,22)(11,19,14,30)(12,27,29,31)(17,35,24,34)(26,33,28,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72286120, #72286158, #72286470, … |
| maximal normal subgroup of | #72286120, #72286158, #72286470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408302);