Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C162 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C8×C16 | C22 |
| commutator | C2 | C162 |
| socle, prime radical | C22 | C8×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(17,18,19,24,20,25,27,36,21,26,28,37,30,38,39,46,22,23,29,33,31,34,40,43,32,35,41,44,42,45,47,48);(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23)(24,33)(25,34)(26,35)(36,43)(37,44)(38,45)(46,48)Polycyclic presentation:
2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bi320 conjugacy classes — view the full character table.
| Frattini subgroup of | #72277905 |
| maximal subgroup of | C16×M64, #72216265, #72216358, … |
| maximal normal subgroup of | C16×M64, #72216265, #72216358, … |
| maximal quotient of | #72225038, C2×#93664, #72234310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 860);