Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1402 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 266 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:
(2,4)(5,6,9,14,10,13,8,12)(7,15,16,11,18,19,17,20)(21,22)(23,26)(24,31,32,27)(25,28)(29,33,35,36)(30,34);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,20)(14,15)(21,23,25,30)(22,26,28,34)(24,29,32,35)(27,33,31,36);(5,8,10,9)(6,12,13,14)(7,16,18,17)(11,19,20,15)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,35)(34,36);(6,13)(11,20)(12,14)(15,19)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101929);