Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417558 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2118 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,22,24,34)(13,17)(14,18)(16,23,32,26)(19,36,38,40)(20,29)(21,30)(25,33)(27,35)(28,37,31,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15,18,30)(12,19,24,38)(13,25,27,20)(16,31,32,28)(17,33,35,29)(22,36,34,40)(23,39,26,37);(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,34)(18,32)(20,39)(21,38)(22,35)(23,27)(25,37)(29,40)(30,31)(33,36);(3,8)(5,7)(13,27)(17,35)(20,25)(22,34)(23,26)(29,33)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=eh,b^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72317977, #72317994, #72318059, … |
| maximal normal subgroup of | #72317977, #72317994, #72318059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356725);