Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C32 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C8×C16 | C22 |
| commutator | C2 | C8×C32 |
| 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 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16);(2,6)(7,12)(8,13)(14,16)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48)Polycyclic presentation:
2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bg320 conjugacy classes — view the full character table.
| Frattini subgroup of | #72278011 |
| maximal subgroup of | #72216220, #72225071, #72226997, … |
| maximal normal subgroup of | #72216220, #72225071, #72226997, … |
| maximal quotient of | #72225071, C2×#93665, #72234310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 861);