Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C82 · Schur multiplier (H2) = C2×C4×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C16 | C22 |
| commutator | C2 | C4×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 402 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 402 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(13,14,16,20,17,21,26,32,18,22,27,33,28,34,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44);(5,6,7,9,8,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,30)(21,31)(22,29)(26,37)(27,35)(28,36)(32,41)(33,39)(34,40)(38,43)(42,44);(1,2,3,4)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,c^b=ci320 conjugacy classes — view the full character table.
| Frattini subgroup of | #72234274 |
| maximal subgroup of | #72210602, #72210619, #72210680, … |
| maximal normal subgroup of | #72210602, #72210619, #72210680, … |
| maximal quotient of | #72234224, #72234226, #72234252 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33715);