Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4328 conjugacy classes of subgroups; only the 610 normal subgroups are shown.
This group has 4328 conjugacy classes of subgroups — too many to draw. Only its 610 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 — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(13,24)(15,26)(17,31)(20,35)(22,25)(27,37)(29,32)(33,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,27)(20,28)(21,26)(22,29)(23,30)(24,31)(25,32)(33,39)(34,37)(35,38)(36,40);(3,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,27)(17,28)(18,26)(22,33)(23,34)(24,35)(25,36)(29,39)(30,37)(31,38)(32,40);(2,6)(5,8)(9,12,14,23)(10,16,18,30)(11,19,21,34)(13,25,24,22)(15,27,26,37)(17,32,31,29)(20,36,35,33)(28,40,38,39);(9,13,14,24)(10,17,18,31)(11,20,21,35)(12,22,23,25)(15,28,26,38)(16,29,30,32)(19,33,34,36)(27,39,37,40)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318059, #72318197, #72318266, … |
| maximal normal subgroup of | #72318059, #72318197, #72318266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340156);