Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1860 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 1860 conjugacy classes of subgroups — too many to draw. Only its 322 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,12,17,10,16)(7,18,20,8,21,23,13,24)(9,25,27,31,28,14,26,30)(19,33,34,36,35,32,29,22);(2,4)(5,7)(6,8)(9,19)(10,13)(11,20)(12,21)(14,22)(15,23)(16,18)(17,24)(25,36)(26,29)(27,34)(28,35)(30,33)(31,32);(5,8)(6,13)(7,15)(9,22)(10,18)(11,23)(12,24)(14,29)(16,21)(17,20)(19,30)(25,34)(26,32)(27,33)(28,36)(31,35);(5,9,11,27,12,28,10,26)(6,14,15,30,17,25,16,31)(7,19,20,34,21,35,13,29)(8,22,23,33,24,36,18,32)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280138, #72313521 |
| maximal normal subgroup of | #72280138, #72313521 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266190);