Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100139 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
798 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,14,16,10)(8,13)(9,12)(11,15,20,17)(18,19)(21,22)(23,26)(24,31,32,27)(25,28)(29,35,36,33)(30,34);(1,2)(3,4)(5,7)(6,10)(8,17)(9,16)(11,19)(12,14)(13,20)(15,18)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(2,4)(5,7)(6,11,12,20)(8,15)(9,16)(10,13,14,19)(17,18)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,g^b=gh,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26661);