Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232393 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1568 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 258 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:
(1,2)(3,4)(5,6,11,15)(7,16,18,19)(8,12,20,17)(9,14,10,13)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,35)(34,36);(5,7,11,18)(6,8,15,20)(9,17,10,12)(13,16,14,19)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(2,4)(5,8,11,20)(6,12,15,17)(7,13,18,14)(9,19,10,16)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,36)(33,34);(5,9,11,10)(6,13,15,14)(7,17,18,12)(8,16,20,19)(23,30)(26,33)(29,36)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139596);