Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C24 |
| commutator | C2 | C22×C82 |
| socle | C22 | C22×C4×C8 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups; only the 678 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 678 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,12,17,13,18,22,30,14,15,23,26,24,27,34,37)(11,16,20,28,21,29,33,38,19,25,31,35,32,36,39,40);(1,2,3,5,4,6,7,8)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(9,11)(10,16)(12,20)(13,21)(14,19)(15,25)(17,28)(18,29)(22,33)(23,31)(24,32)(26,35)(27,36)(30,38)(34,39)(37,40);(11,19)(16,25)(20,31)(21,32)(28,35)(29,36)(33,39)(38,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,g^2=i,b^a=bi,d^c=di272 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#3407, #72231854, #72231856, … |
| maximal normal subgroup of | C8×#3407, #72231854, #72231856, … |
| maximal quotient of | #72226970, #72226971, #72226973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387095);