Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1218 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)(7,15,18,31)(8,19,21,32)(9,23,26,13)(10,14)(11,17)(12,22)(16,27)(20,29)(24,28)(25,30)(33,35,36,34);(1,2)(3,4)(5,7,10,18)(6,11,14,28)(8,20,21,16)(9,24,26,17)(12,29,30,27)(13,31,23,15)(19,36,32,33)(22,34,25,35);(2,4)(8,21)(12,30)(13,23)(15,31)(16,20)(22,25)(27,29)(34,35);(5,8,10,21)(6,12,14,30)(7,16,18,20)(9,25,26,22)(11,27,28,29)(13,32,23,19)(15,33,31,36)(17,35,24,34)Polycyclic presentation:
2^9|a^2=f,b^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 | #72285425, #72286038, #72286125, … |
| maximal normal subgroup of | #72285425, #72286038, #72286125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408301);